E7 TCR T Cell Induction Immunotherapy for Stage IIB-IVA Cervical Cancer

-INCLUSION CRITERIA: 1. Participants with histologically or cytologically confirmed carcinoma of the cervix that has not been treated, with clinical staging as follows: * Lead-in safety cohort: FIGO (International Federation of Gynecology and Obstetrics) stage IIIC-IVA (2018 International FIGO Staging System) * After lead-in safety cohort: FIGO stage IIB-IVA (2018 International FIGO Staging System) 2. Human papillomavirus 16 (HPV16+) tumor and HLA-A\*02:01+ Human leukocyte antigen (HLA) type. Note: HLA-A\*02 is also acceptable for enrollment but not for treatment. 3. Measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 criteria or Positron Emission Tomography Response Criteria in Solid Tumors (PERCIST) (if not eligible by RECIST 1.1). 4. Age 18 years. Because no dosing or adverse event data are currently available on the use of E7 TCR T cells in participants \<18 years of age, children are excluded from this study. Note: This age range is consistent with the age of participants with the disease being studied. 5. Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1. 6. Women of child-bearing potential must have a negative pregnancy test because E7 TCR T cells have unknown potential for teratogenic or abortifacient effects. Women of child- bearing potential are defined as all women who are not post-menopausal or who have not had a hysterectomy. Note: Postmenopausal will be defined in this study as women over the age of 55 who have not had a menstrual period in at least 1 year. 7. The effects of E7 TCR T cells on the developing human fetus are unknown. For this reason and because the chemotherapy agents used in this trial are known to be teratogenic, women of child-bearing potential must agree to use adequate contraception (e.g., intrauterine device, hormonal or barrier method of birth control; abstinence; tubal ligation or vasectomy) prior to study entry and for four months after treatment. Should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately. 8. Seronegative for HIV (human immunodeficiency virus) antibody. The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants who are HIV seropositive can have decreased immune-competence and thus be less responsive to the experimental treatment. 9. Seronegative for hepatitis B antigen and hepatitis C antibody. If hepatitis C antibody test is positive, then the participant 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. 10. Must be willing to participate in Gene Therapy Long Term Follow up Protocol (20C0051), which will follow participants for up to 15 years per Food and Drug Administration (FDA) requirements. 11. Participants must have organ and marrow function as defined below: * leukocytes \>=3,000/mcL * absolute neutrophil count \>=1,500/mcL * platelets \>=100,000/mcL * hemoglobin \>=9.0 g/dL * total bilirubin within normal institutional limits except in participants with Gilbert's Syndrome who must have a total bilirubin \< 3.0 mg/dL * Aspartate aminotransferase (AST) serum glutamic oxaloacetic transaminase (SGOT)/alanine aminotransferase (ALT) serum glutamate-pyruvate transaminase (SGPT) Serum ALT/AST \< 2.5X ULN * creatinine clearance Calculated creatinine clearance (CrCl) \>=50 mL/min/1.73 m\^2 for participants with creatinine levels above institutional normal (by the Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation) 12. Ability of subject to understand and the willingness to sign a written informed consent document. EXCLUSION CRITERIA: 1. Previous treatment for invasive cervical cancer including: * Chemotherapy or other systemic treatments * Radiation therapy * Hysterectomy (prior LEEP procedure or cone biopsy is allowed) 2. Participants who are receiving any other investigational agents. 3. History of severe allergic reactions attributed to compounds of similar chemical or biologic composition to agents used in study. 4. Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations at the time of treatment that would limit compliance with study requirements. 5. Because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with E7 TCR T cells, breastfeeding should be discontinued if the mother is treated with E7 TCR T cells. These potential risks may also apply to other agents used in this study. 6. Participants with any form of systemic immunodeficiency, including acquired deficiency such as HIV or primary immunodeficiency such as Severe Combined Immunodeficiency Disease, are ineligible. The experimental treatment being evaluated in this protocol depends on an intact immune system. Participants who have decreased immune competence may be less responsive to the treatment. 7. Participants on immunosuppressive drugs including corticosteroids. 8. Participants with autoimmune diseases such as Crohn's disease, ulcerative colitis, rheumatoid arthritis, autoimmune hepatitis, autoimmune pancreatitis, or systemic lupus erythematosus. Hypothyroidism, vitiligo and other minor autoimmune disorders are not exclusionary. 9. Participants with a second invasive malignancy requiring treatment within the last 2 years are not eligible with the following exceptions: * Ductal carcinoma in situ (DCIS) of the breast * Cutaneous skin cancers requiring only local excision