Interleukin-15 and -21 Armored Glypican-3-specific Chimeric Antigen Receptor Expressed in T Cells for Pediatric Solid Tumors

Procurement Eligibility Inclusion Criteria: * Diagnosis of GPC3-positive\* solid tumors (as determined by immunohistochemistry with an extent score of \>=Grade 2 \[\>25% positive tumor cells\] and an intensity score of \>= 2 \[scale 0-4\]). * Age ≥1 year and ≤ 21 years * Lansky or Karnofsky score ≥60% * Life expectancy ≥16 weeks * Barcelona Clinic Liver Cancer Stage A, B or C (for patients with hepatocellular carcinoma only) * Child-Pugh-Turcotte score \<7 (for patients with hepatocellular carcinoma only) * Informed consent explained to, understood by and signed by patient/guardian. Patient/guardian given copy of informed consent Exclusion Criteria: * History of hypersensitivity reactions to murine protein-containing products OR presence of human anti-mouse antibody (HAMA) prior to enrollment (only patients who have received prior therapy with murine antibodies). * History of organ transplantation * Known HIV positivity * Active bacterial, fungal or viral infection (except Hepatitis B or Hepatitis C virus infections) Treatment Eligibility Inclusion Criteria: * Age ≥ 1 year and ≤ 21 years * Barcelona Clinic Liver Cancer Stage A, B or C (for patients with hepatocellular carcinoma only) * Lansky or Karnofsky score ≥ 60% * Child-Pugh-Turcotte score \< 7 (for patients with hepatocellular carcinoma only) * Adequate organ function: * Creatinine clearance as estimated by Cockcroft Gault or Schwartz ≥ 60 ml/min * Total bilirubin \< 3 times ULN for age * INR ≤1.7 (for patients with hepatocellular carcinoma only) * Absolute neutrophil count \> 750/µl * Platelet count \> 75,000/µl (Needs to be confirmed prior to treatment whether with or without transfusion) * Hgb ≥ 8.0 g/dl (Needs to be confirmed prior to treatment whether with or without transfusion) * Pulse oximetry ≥ 92% on room air * Incurable disease after treatment with up- front therapy (Patients who have relapsed disease despite a standard of care salvage therapy) * Wash out period, such that patient has recovered from acute toxic effects of all prior chemotherapy and investigational agents before entering this study, and returned to their clinical baseline, as determined by history and physical exam. * Sexually active patients must be willing to utilize one of the more effective birth control methods for 6 months after the T-cell infusion. * Informed consent explained to, understood by and signed by patient/guardian. Patient/guardian given copy of informed consent Exclusion Criteria: * Pregnancy or lactation * Uncontrolled infection * Systemic steroid treatment (greater than or equal to 0.5 mg prednisone equivalent/kg/day, dose adjustment or discontinuation of medication must occur at least 24 hours prior to CAR T cell infusion) * Known HIV positivity * Active bacterial, fungal or viral infection \[except Hepatitis B (HBV patients with active disease who meet the criteria for anti-HBV therapy should be on a suppressive antiviral therapy prior to initiation of cancer therapy) or Hepatitis C virus infections (should have completed curative antiviral treatment with HCV viral load below the limit of quantification\] * Congestive heart failure (as defined by New York Heart Association Functional Classification III or IV), unstable angina, serious uncontrolled cardiac arrhythmia, a myocardial infarction within 6 months prior to study entry or a history of myocarditis * Active autoimmune or inflammatory disorder * Live vaccines within 30 days prior to enrollment * History of organ transplantation * History of hypersensitivity reactions to murine protein-containing products OR presence of human anti-mouse antibody (HAMA) prior to enrollment (only patients who have received prior therapy with murine antibodies)