Autologous CD8+ and CD4+ Transgenic T Cells Expressing High Affinity KRASG12V Mutation-Specific T Cell Receptors (FH-A11KRASG12V-TCR) in Treating Patients With Metastatic Solid Tumor Cancers With KRAS G12V Mutations

Inclusion Criteria: * LEUKAPHERESIS: Diagnosis of metastatic solid tumor * LEUKAPHERESIS: Tissue confirmation of solid tumor. Confirmation of diagnosis must be or have been performed by internal pathology review of archival biopsy material or other pathologic material at Fred Hutch/University of Washington Cancer Consortium (UWMC) * LEUKAPHERESIS: HLA-A\*11:01 confirmed through HLA typing at a clinically accredited laboratory * LEUKAPHERESIS: Previously documented KRASG12V mutation in tumor or plasma cell-free deoxyribonucleic acid (cfDNA) specimens by polymerase chain reaction (PCR) or next-generation sequencing (NGS) test * LEUKAPHERESIS: Measurable disease by RECIST 1.1 criteria: Participants must have measurable disease, defined as at least one target lesion that can be measured in at least one dimension (longest diameter to be recorded) as ≥ 10 mm, unless lymph node in which case short axis must be ≥ 15 mm. Baseline imaging (for example, diagnostic CT chest/abdomen/pelvis and imaging of the affected extremity as appropriate), brain imaging (MRI or CT scan) if clinically indicated, must be obtained within 8 weeks of the first planned T cell infusion. MRI can be substituted for CT in participants unable to have CT contrast * LEUKAPHERESIS: Participants must be willing to undergo tumor biopsy for research purposes if safe and feasible at baseline (prior to first T cell infusion), 2-3 weeks after the first T cell infusion, and approximately 2 weeks +/- 1 week after the 2nd …