The goal of this clinical research study is to find the highest tolerable dose of the combination of alisertib and TAK-228 that can be given to participants with advanced solid tumors that are associated with HPV. Researchers also want to learn if the study drug combination can help to control advanced solid tumors.

Study Groups:

If participant is found to be eligible to take part in this study, they will be assigned to a dose level based on when they join the study. Up to 5 dose levels of alisertib and TAK-228 will be tested. The first group of participants will receive the lowest dose level. Each new group will receive a higher dose than the group before it, if no intolerable side effects were seen. This will continue until the highest tolerable dose of alisertib and TAK-228 is found.

Up to 14 participants will be enrolled onto a Dose Expansion group after the highest tolerable dose combination has been found.

Study Drug Administration:

Each cycle is 21 days.

On Days 1-7 of each cycle, participant will take alisertib by mouth 2 times a day with a glass of water (about 4 ounces).

On Days 3-18 of each cycle, participant will take TAK-228 by mouth 1 time a day Participant should not eat or drink anything for 2 hours before and 1 hour after taking TAK-228.

Study Visits:

Participant will need to take their blood sugar levels every morning while they are on study. Participant will be given a blood sugar monitor and the study staff will show them how to use it. Participant will also have their blood sugar tested in the clinic Participant must fast for at least 8 hours before each of these blood sugar tests. If the doctor thinks it is in participant's best interest, this testing may be decreased to 1 time a week after the first 2 months.

During Week 1 of Cycle 1:

• Participant will have a physical exam.
• Blood (about 3 teaspoons) will be drawn for routine tests and to test blood clotting, participant's blood fat levels, and their blood sugar levels.
• Urine will be collected for routine tests

On Day 1 of Cycles 1 and 2, you will have an EKG.

One (1) time during Weeks 2 and 3 of Cycle 1, blood (about 2 teaspoons) for routine tests and to test participant's blood sugar.

One (1) time a week during Cycles 1 and 2, blood (about 2 teaspoons) will be drawn for routine tests.

During Week 1 of Cycle 2 and beyond:

• Participant will have a physical exam.
• Urine will be collected for routine tests.

During Week 1 of Cycle 3 and beyond, blood (about 4 teaspoons) will be drawn routine tests and to test blood clotting, participant's blood fat levels, and their blood sugar levels.

During Cycles 2, 4, and 6 and then every 3 cycles, participant will have a CT, MRI, or positron emission tomography (PET) scan to check the status of the disease.

Length of Study:

Participant may continue taking the study drugs for as long as they are benefitting. Participant will no longer be able to take the study drugs if the disease gets worse, if intolerable side effects occur, if the doctor thinks it is in their best interest, if the study is stopped, if they need treatment that is not allowed on this study, or if they are unable to follow study directions.

Patient's participation on the study will be over after the end-of-study visit.

End of Study Visit:

Within 30 days of participant's last dose of study drugs:

• Participant will have an EKG to check your heart function.
• Participant will have a physical exam.
• Urine will be collected for routine tests.
• Blood (about 3 teaspoons) will be drawn for routine tests and to test blood clotting, participant's blood fat levels, and their blood sugar levels.
• Participant will have a CT, MRI, or PET scan to check the status of the disease.

This is an investigational study. Alisertib and TAK-228 are not FDA approved or commercially available. They are currently being used for research purposes only.

The study doctor can explain how the study drugs are designed to work.

Up to 44 participants will take part in this study. All will be enrolled at MD Anderson.

Inclusion Criteria:

1. Patients with locally advanced or metastatic HPV associate malignancy (cervical, vaginal, vulvar, penile, anal or oropharyngeal carcinoma), either refractory to standard therapy or for which no effective standard therapy that confers clinical benefit is available.
2. Patients must have measurable disease, as defined by RECIST 1.1.
3. Male or female patients 18 years or older.
4. Eastern Cooperative Oncology Group (ECOG) performance status of 0 to 2.
5. For women: - Postmenopausal for at least 1 year before the screening visit, OR - Surgically sterile, OR - If they are of childbearing potential, agree to practice 1 effective methods of contraception and 1 additional effective (barrier) method, at the same time, from the time of signing the informed consent through 90 days (or longer, as mandated by local labeling [eg. USPI, SmPC, etc;])after the last dose of study drug OR agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the patient (Periodic abstinence [e.g, calendar, ovulation, symptothermal, postovulation methods] and withdrawal, spermicides only and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.)
6. For men, even if surgically sterilized (ie, status post-vasectomy), they must: - Agree to practice highly effective barrier contraception during the entire study treatment period and through 120 days after the last dose of study drug, OR agree to practice true abstinence, when this is in line with the preferred and usual lifestyle of the patient (Periodic abstinence [e.g, calendar, ovulation, symptothermal, postovulation methods for the female partner] and withdrawal, spermicides only, and lactational amenorrhea are not acceptable methods of contraception. Female and male condoms should not be used together.) - Agree not to donate sperm during the course of this study or 120 days after receiving their last dose of study drug
7. Screening clinical laboratory values as specified: -Bone marrow reserve consistent with: absolute neutrophil count (ANC) >/= 1.5 x 10^9/L; platelet count >/= 100 x 10^9/L; hemoglobin >/= 9 g/dL without transfusion within 1 week preceding study drug administration. -Hepatic: total bilirubin </= 1.5 x upper limit of normal (ULN), transaminases (aspartate aminotransferase/serum glutamic oxaloacetic transaminase-AST/SGOT and alanine aminotransferase/serum glutamic pyruvic transaminase-ALT/SGPT) </= 2.5 x ULN (</= 5 x ULN if liver metastases are present); -Renal: creatinine clearance >/=50 mL/min based either on Cockcroft-Gault estimate or based on urine collection (12 or 24 hour); -Metabolic: Glycosylated hemoglobin (HbA1c)<7.0%, fasting serum glucose (</= 130 mg/dL) and fasting triglycerides </= 300 mg/dL;
8. Left ventricular ejection fraction (LVEF) be at least >/= institutional standard of normal as measured by echocardiogram (ECHO) or multiple gated acquisition scan (MUGA) within 4 weeks prior to first study drug administration
9. Ability to swallow oral medications.
10. Voluntary written consent must be given before performance of any study related procedure not part of standard medical care, with the understanding that consent may be withdrawn by the patient at any time without prejudice to future medical care.
11. Patients who have a history of brain metastasis are eligible for the study provided that all the following criteria are met:-Brain metastases which have been treated, -No evidence of disease progression for >/= 3 months or hemorrhage after treatment, -Off-treatment with dexamethasone for 4 weeks before administration of the first dosing, -No ongoing requirement for dexamethasone or anti-epileptic drugs

Exclusion Criteria:

1. Other clinically significant co-morbidities, such as uncontrolled pulmonary disease, active central nervous system disease, active infection, or any other condition that could compromise the patient's participation in the study.
2. Known human immunodeficiency virus infection.
3. Known hepatitis B (they test positive for the presence of at least one of the following three markers in blood (to be evaluated at screening): hepatitis B surface antigen (HBsAG), antibodies against hepatitis B core antigen (anti-HBc), or hepatitis B viral load (HBV DNA), or known active hepatitis C infection.
4. Any serious medical or psychiatric illness that could, in the investigator's opinion, potentially interfere with the completion of treatment according to this protocol.
5. Diagnosed or treated for another malignancy within 2 years before administration of the first dose of study drug, or previously diagnosed with another malignancy and have any evidence of residual disease. Patients with nonmelanoma skin cancer or carcinoma in situ of any type are not excluded if they have undergone complete resection.
6. Female subject who is pregnant or breast-feeding. Confirmation that the subject is not pregnant must be established by a negative serum b-human chorionic gonadotropin (b-hCG) pregnancy test result obtained during screening. Pregnancy testing is not required for post-menopausal or surgically sterilized women.
7. Inability or unwillingness to swallow oral medication. Manifestations of malabsorption due to prior gastrointestinal (GI) surgery, GI disease, or for an unknown reason that may alter the absorption. In addition, patients with enteric stomata are also excluded.
8. Therapy with any investigational products within 21 days before the first dose of study drug.
9. History of any of the following within the last 6 months before administration of the first dose of the drug:-Ischemic myocardial event, including angina requiring therapy and artery revascularization procedures, -Ischemic cerebrovascular event, including transient ischemic attack and artery revascularization procedures, -Requirement for inotropic support (excluding digoxin) or serious (uncontrolled) cardiac arrhythmia (including atrial flutter/fibrillation, ventricular fibrillation or ventricular tachycardia), -Placement of a pacemaker for control of rhythm, -New York Heart Association (NYHA) Class III or IV heart failure, -Pulmonary embolism
10. Significant active cardiovascular or pulmonary disease including: -Uncontrolled hypertension (i.e., systolic blood pressure >180 mm Hg, diastolic blood pressure > 95 mm Hg). Use of anti-hypertensive agents to control hypertension before Cycle1 Day 1 is allowed., -Pulmonary hypertension, -Uncontrolled asthma or O2 saturation < 90% by arterial blood gas analysis or pulse oximetry on room air, -Significant valvular disease; severe regurgitation or stenosis by imaging independent of symptom control with medical intervention, or history of valve replacement, -Medically significant (symptomatic) bradycardia, -History of arrhythmia requiring an implantable cardiac defibrillator, -Baseline prolongation of the rate-corrected QT interval (QTc) (e.g., repeated demonstration of QTc interval > 480 milliseconds, or history of congenital long QT syndrome, or torsades de pointes)
11. Poorly controlled diabetes mellitus defined as glycosylated hemoglobin (HbA1c) > 7% ; patients with a history of transient glucose intolerance due to corticosteroid administration may be enrolled in this study if all other inclusion/exclusion criteria are met.
12. Treatment with strong inhibitors and/or inducers of cytochrome P450 (CYP) 3A4, CYP2C9 or CYP2C19 within 1 week preceding the first dose of study drug.
13. Receipt of corticosteroids within 7 days prior to the first dose of study treatment, unless patient has been taking a continuous dose of no more than 15 mg/day of prednisone or equivalent for at least 1 month prior to first dose of study treatment. Low dose steroid use for the control of nausea and vomiting, topical steroid use and inhaled steroids are permitted.
14. The intermittent use of PPI, H2-antagonists and antacids (including carafate) is only allowed within these guidelines: -PPI until D-5 prior to the first dose of alisertib and prohibited for the duration of the study, -H2 antagonists until D-1 and after the dosing of alisertib is done, -Antacid formulations until 2 hours before dosing and after 2 hours following dosing
15. Radiation therapy to more than 25% of the bone marrow. Whole pelvic radiation is considered to be over 25%.
16. Prior allogeneic bone marrow or organ transplantation.
17. Systemic infection requiring IV antibiotic therapy within 14 days preceding the first dose of study drug, or other severe infection.
18. Serious medical or psychiatric illness likely to interfere with participation in this clinical study. Other severe acute or chronic medical or psychiatric condition, including uncontrolled diabetes, malabsorption, resection of the pancreas or upper small bowel, requirement for pancreatic enzymes, any condition that would modify small bowel absorption of oral medications, or laboratory abnormality that may increase the risk associated with study participation or investigational product administration or may interfere with the interpretation of study results and, in the judgment of the investigator, would make the patient inappropriate for enrollment in this study.
19. Administration of myeloid growth factors or platelet transfusion within 14 days prior to the first dose of study treatment.
20. Prior administration of an Aurora A kinase-targeted agent, including alisertib
21. Known history of uncontrolled sleep apnea syndrome and other conditions that could result in excessive daytime sleepiness, such as severe chronic obstructive pulmonary disease; requirement for supplemental oxygen.
22. Requirement for constant administration of H2 antagonist.