A Study of Gilteritinib (ASP2215) Combined With Chemotherapy in Children, Adolescents and Young Adults With FMS-like Tyrosine Kinase 3 (FLT3)/Internal Tandem Duplication (ITD) Positive Relapsed or Refractory Acute Myeloid Leukemia (AML)

Inclusion Criteria: * Subject is aged ≥ 6 months and \< 21 years of age\* at the time of signing informed consent and/or assent, as applicable. * \*For phase 2: Enrollment of subjects from 6 months to less than 1 year and 1 year to less than 2 years will be dependent on the establishment of recommended phase 2 dose (RP2D) in the respective age groups during phase 1. * Subject has a diagnosis of acute myeloid leukemia (AML) according to The French-American-British (FAB) classification with ≥ 5% blasts in the bone marrow, with or without extramedullary disease (except subjects with active central nervous system \[CNS\] leukemia). * In the phase 1 portion of the study, subject must be in first or greater relapse or refractory to induction therapy with no more than 1 attempt at remission induction (up to 2 induction cycles). * For the phase 2 portion of the study, subject must be in refractory to or at the first hematologic relapse after first-line remission induction AML therapy (up to 2 induction cycles). * Subject has fully recovered from the acute toxic effects of all prior chemotherapy, immunotherapy, or radiotherapy prior to entering this study. * Myelosuppressive chemotherapy: * For subject who relapses while receiving cytotoxic therapy, at least 21 days must have elapsed since the completion of cytotoxic therapy and prior to screening, unless the subject has recovered earlier than 21 days. * Cytoreduction with the following can be initiated and continued for up to 24 hours prior to the start of systemic protocol therapy (cycle 1 day -1). * hydroxyurea, * low dose cytarabine (100 mg/m\^2 per dose once daily for 5 days) or * other low dose/maintenance therapies as per local site practice. * Subject who has received other FLT3 inhibitors (e.g., lestaurtinib, sorafenib, etc) is eligible for this study. * Hematopoietic growth factors: at least 7 days must have elapsed since the completion of therapy with a growth factor and prior to screening. * Biologic (anti-neoplastic agent): at least 7 days must have elapsed since the completion of therapy with a biologic agent and prior to screening. For agents that have known adverse events (AEs) occurring beyond 7 days after administration, this period must be extended beyond the time during which AEs are known to occur. * X-ray treatment (XRT): * 14 days must have elapsed for local palliative XRT for CNS chloromas and prior to screening; no washout period is necessary for other chloromas; * Prior to screening, 90 days must have elapsed if the subject had a prior traumatic brain injury or has received craniospinal XRT. * For subject undergoing hematopoietic stem cell transplant (HSCT), at least 90 days must have elapsed since HSCT and subject must not have active graft-versus-host disease (GVHD). * Subject has Karnofsky score ≥ 50 (if the subject is of ≥ 16 years of age) or Lansky score of ≥ 50 (if the subject is \< 16 years of age). A score \< 50 is acceptable if related to the subject's leukemia. * Subject must meet the following criteria as indicated on the clinical laboratory tests. * Serum aspartate aminotransferase (AST) and alanine aminotransferase (ALT) ≤ 3.0 x upper limit normal (ULN) for age * Total serum bilirubin ≤ 1.5 x ULN for age * Estimated glomerular filtration rate of \> 60 mL/min/1.73 m\^2. * A female subject is eligible to participate if she is not pregnant and at least 1 of the following conditions applies: * Not a woman of childbearing potential (WOCBP) OR * WOCBP who agrees to follow the contraceptive guidance throughout the treatment period and for at least 180 days after the final study drug administration. * Female subject must agree not to breastfeed starting at Screening, and throughout the study period and for 60 days after the final study drug administration. * Female subject must not donate ova starting at Screening and throughout the study, and for 180 days after the final study drug administration. * A male subject with female partner(s) of childbearing potential must agree to use contraception during the treatment period and for at least 180 days after the final study drug administration. * A male subject must not donate sperm during the treatment period and for at least 120 days after the final study drug administration. * Male subject with a pregnant or breastfeeding partner(s) must agree to remain abstinent or use a condom for the duration of the pregnancy or time partner is breastfeeding throughout the study period and for 180 days after the final study drug administration. * Subject and subject's parent(s) or legal guardian agrees not to participate in another interventional study while on treatment. * Live Vaccines - At least 6 weeks must have elapsed since the administration of the last dose of a live vaccine and prior to the initiation of study treatment (cycle 1, day -1) * Phase 1: Subject is positive for FLT3 (ITD and/or tyrosine kinase domain \[TKD\]) mutation in bone marrow or blood as determined by the local institution. * Phase 2: Subject is positive for the FLT3 (ITD) mutation in bone marrow or blood as determined by the local institution. Exclusion Criteria: * Subject has active CNS leukemia. * Subject has uncontrolled or significant cardiovascular disease, including: * Diagnosed or suspected congenital long QT syndrome or any history of clinically significant ventricular arrhythmias (such as ventricular tachycardia, ventricular fibrillation, or Torsades de Pointes (TdP)); any history of arrhythmia will be discussed with the sponsor prior to subject's entry into the study * Prolonged Fridericia's Correction Formula (QTcF) interval on pre-entry electrocardiogram (ECG) (≥ 450 ms) * Any history of second or third degree heart block (may be eligible if the subject currently has a pacemaker) * Heart rate \< 50 beats/minute on pre-entry ECG * Uncontrolled hypertension * Complete left bundle branch block * Subject has systemic fungal, bacterial, viral or other infection that is exhibiting ongoing signs/symptoms related to the infection without improvement despite appropriate antibiotics or other treatment. The subject needs to be off pressors and have negative blood cultures for 48 hours. * Subject is receiving or plans to receive concomitant chemotherapy, radiation therapy, or immunotherapy other than as specified in the protocol. * Subject has active clinically significant GVHD or is on treatment with immunosuppressive drugs for treatment of active GVHD, with the exception of subjects being weaned from systemic corticosteroids where the subject is receiving ≤ 0.5 mg/kg of prednisone (or equivalent) daily dose for prior GVHD. Subject has received calcineurin inhibitors within 4 weeks prior to screening, unless used as GVHD prophylaxis. * Subject has active malignant tumors other than AML. * Subject has any significant concurrent disease, illness, psychiatric disorder or social issue that would compromise subject safety or compliance; interfere with consent, study participation, follow-up or interpretation of study results. * Subject has hypokalemia and/or hypomagnesemia at Screening (defined as values below institutional lower limit of normal \[LLN\]). Repletion of potassium and magnesium levels during the screening period is allowed. * Subject requires treatment with concomitant drugs that are strong inducers of cytochrome P450 (CYP)3A/P-glycoprotein (P-gp). * Subject is known to have human immunodeficiency virus infection. * Subject has active hepatitis B or C, or other active hepatic disorder. * Subjects with positive hepatitis B surface antigen (HBsAg) or detectable hepatitis B DNA are not eligible. * Subjects with negative HBsAg, positive hepatitis B core antibody and negative hepatitis B surface antibody will be eligible if hepatitis B DNA is undetectable. * Subjects with antibodies to hepatitis C virus will be eligible if hepatitis C RNA is undetectable. * Subject must wait for at least 5 half-lives after stopping therapy with any investigational agent and before starting gilteritinib. * Subject has a known or suspected hypersensitivity to gilteritinib, cytarabine, fludarabine, granulocyte colony-stimulating factor (G-CSF) or any components of the formulation used.