E7 TCR T Cells for Human Papillomavirus-Associated Cancers

* INCLUSION CRITERIA: 1. Measurable metastatic or refractory/recurrent human papillomavirus (HPV-16+ cancer (determined by in situ hybridization (ISH) or a polymerase chain reaction (PCR)-based test). 2. Patients must be human leukocyte antigen (HLA-A\*02 by low resolution typing, and HLA-A\*02:01 by one of the high-resolution type results. 3. All patients must have received prior first line standard therapy or declined standard therapy. 4. Patients with three or fewer brain metastases that have been treated with surgery or stereotactic radiosurgery are eligible. Lesions that have been treated with stereotactic radiosurgery must be clinically stable for one month before protocol treatment. Patients with surgically resected brain metastases are eligible. 5. Greater than or equal to 18 years of age. 6. Able to understand and sign the Informed Consent Document. 7. Clinical performance status of Eastern Cooperative Oncology Group (ECOG) 0 or 1. 8. Individuals must be willing to practice birth control from the time of enrollment on this study up to twelve (12) months after treatment. Individuals must be willing to undergo testing for HPV-16 prior to becoming pregnant after this period. 9. Individuals of childbearing potential must have a negative pregnancy test because of the potentially dangerous effects of the treatment on the fetus. Individuals of childbearing potential are defined as all individuals except individuals who are postmenopausal or who have had a hysterectomy. Postmenopausal will be defined as individuals over the age of 55 who have not had a menstrual period in at least one year. Because there is a potential risk for adverse events in nursing infant's secondary to treatment of the mother with E7 T cell receptor (TCR) transduced peripheral blood lymphocytes (PBLs), breastfeeding should be discontinued if the individual is treated with E7 TCR transduced PBL. These potential risks may also apply to other agents used in this study. 10. Serology: * Seronegative for human immunodeficiency virus (HIV) antibody. (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who are HIV seropositive can have decreased immune-competence and thus are less responsive to the experimental treatment and more susceptible to its toxicities.) * Seronegative for hepatitis B antigen, and seronegative for hepatitis C antibody. If hepatitis C antibody test is positive, then the patient must be tested for the presence of antigen by reverse transcription-polymerase chain reaction (RT-PCR) and be hepatitis C virus (HCV) ribonucleic acid (RNA) negative. a. Hematology: * Absolute neutrophil count greater than 1000/mm\^3 without the support of filgrastim. * White blood count (WBC) greater than or equal to 3000/mm\^3 * Platelet count greater than or equal to 100,000/mm\^3 * Hemoglobin \> 8.0 g/dL b. Chemistry: * Serum Alanine aminotransferase (ALT)/Aspartate aminotransferase (AST) less than or equal to 2.5 times the upper limit of normal * Calculated creatinine clearance (CCr) greater than or equal to 50 mL/min/1.73\^2 using the Cockcroft-Gault equation * Total bilirubin less than or equal to 1.5 mg/dL, except in patients with Gilbert's Syndrome who must have a total bilirubin less than 3.0 mg/dL c. More than four weeks must have elapsed since any prior systemic therapy at the time the patient receives the E7 TCR cells. Note: Patients may have undergone minor surgical procedures within the past three weeks, as long as all toxicities have recovered to Grade 1 or less. EXCLUSION CRITERIA: 1. Active systemic infections (for e.g.: requiring anti-infective treatment), coagulation disorders or other active major medical illnesses of the cardiovascular, respiratory or immune system, as evidenced by a positive stress thallium or comparable test, myocardial infarction, cardiac arrhythmias, severe obstructive or restrictive pulmonary disease. Patients with abnormal pulmonary function tests but stable obstructive or restrictive pulmonary disease may be eligible. 2. Any form of primary immunodeficiency (such as Severe Combined Immunodeficiency Disease). 3. Concurrent opportunistic infections (The experimental treatment being evaluated in this protocol depends on an intact immune system. Patients who have decreased immune competence may be less responsive to the experimental treatment and more susceptible to its toxicities). 4. Patients with autoimmune diseases such as Crohn's disease, ulcerative colitis, rheumatoid arthritis, autoimmune hepatitis or pancreatitis, and systemic lupus erythematosus. Hypothyroidism, vitiligo and other minor autoimmune disorders are not exclusionary. 5. Patients on immunosuppressive drugs including corticosteroids. With the exception of: intranasal, inhaled, topical steroids, or local steroid injection (e.g., intra-articular injection) -Systemic corticosteroids at physiologic doses 10 mg/day of prednisone or equivalent; or, -Steroids as premedication for hypersensitivity reactions (e.g., computed tomography (CT) scan premedication) 6. History of severe immediate hypersensitivity reaction to cyclophosphamide, fludarabine or aldesleukin. 7. Patients with a history of coronary revascularization or ischemic symptoms unless patient has a normal cardiac stress test. 8. Documented left ventricular ejection fraction (LVEF) of less than or equal to 45% tested. The following patients will undergo cardiac evaluations 1. Clinically significant atrial and/or ventricular arrhythmias including but not limited to: atrial fibrillation, ventricular tachycardia, second or third degree heart block or 2. Age greater than or equal to 50 years old 9. Any other condition, which would, in the opinion of the Principal Investigator, indicate that the subject is a poor candidate for the clinical trial or would jeopardize the subject or the integrity of the data obtained. 10. Subjects with baseline screening pulse oxygen level of \< 95% on room air will not be eligible. If the underlying cause of hypoxia improves, then they may be reevaluated