Evaluation of LY2606368 Therapy in Combination With Cyclophosphamide or Gemcitabine for Children and Adolescents With Refractory or Recurrent Group 3/Group 4 or SHH Medulloblastoma Brain Tumors

Inclusion Criteria: Screening Phase * Participants with recurrent, refractory, or progressive medulloblastoma. * Age ≥ 1 year and \< 25 years at the time of screening. * Participants and/or guardian can understand and is willing to sign a written informed consent document according to institutional guidelines. Exclusion Criteria: Screening Phase * Previous exposure to any CHK1 inhibitor. * Participants with a history of clinically significant, uncontrolled heart disease and/or repolarization abnormalities. * Participants with any history of QTc prolongation (i.e. QTc interval of \> 480 msec). Inclusion Criteria: Strata A and B * Participant must be ≥1 year and \<25 years of age at time of screening. * Participant must have recurrent, progressive or refractory Group 3/Group 4 or SHH medulloblastoma (per central pathology confirmation of primary tissue and/or relapsed tissue). Central pathology review previously completed at St. Jude Children's Research Hospital using equivalent methods can be used for enrollment. Note: Group 3/Group 4 may be referred to as Non-WNT Non-SHH (NWNS) in pathology reports. Medulloblastoma patients with indeterminate molecular subgroup after central pathology review are eligible for enrollment on stratum A. * Participant must have measurable or evaluable disease as defined in the protocol. * Participant must have received their last dose of myelosuppressive anticancer chemotherapy at least 3 weeks prior to study enrollment. * Participants must have had their last fraction of radiation (including CSI) at least 4 weeks prior to study enrollment. Participants who received radiation therapy for palliation must have had their last fraction of radiation at least 2 weeks prior to study enrollment. \-- Note: Participants must have relapsed with recurrent, progressive or refractory disease after any prior radiation therapy that is not considered palliative. Palliative radiation therapy is defined as local small port RT to alleviate and/or palliate symptoms. (CSI, whole brain RT, large field/port RT, or large field/port multilevel spinal RT will not be considered palliative at any dose.) * Participant who are receiving corticosteroids must be on a stable or decreasing dose for at least 1 week prior to enrollment with no plans for escalation. * Participant must have a Lansky (≤ 16 years) or Karnofsky (\> 16 years) performance score of ≥50 and, in the opinion of the investigator, a minimum life expectancy of at least 6 weeks. \-- Note: Participants who are unable to walk because of paralysis, but who are up in a wheelchair, will be considered ambulatory for the purpose of assessing the performance score. * Participant must have adequate bone marrow and organ function as defined as: * ANC ≥ 1.0 x 10\^9/L without growth factor support within 7 days * Platelet count ≥ 75x 10\^9/L without support of a platelet transfusion within 7 days * Hemoglobin ≥8.0 g/dL without support of a blood transfusion within 7 days * Potassium, total calcium (corrected for serum albumin), magnesium, sodium and phosphorus within institutional normal limits or corrected to within normal limits with supplements before first dose of study medication * Serum creatinine ≤ the maximum serum creatinine based on age/gender: Age: 1 to \< 2 years; maximum serum creatinine (mg/dL): 0.6 (male, female); Age: 2 to \< 6 years; maximum serum creatinine (mg/dL): 0.8 (male, female); Age: 6 to \< 10 years; maximum serum creatinine (mg/dL): 1 (male, female); Age: 10 to \< 13 years; maximum serum creatinine (mg/dL): 1.2 (male, female); Age: 13 to \< 16 years; maximum serum creatinine (mg/dL): 1.5 (male), 1.4 (female); Age :≥ 16 years; maximum serum creatinine (mg/dL): 1.7 (male), 1.4 (female). * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 2.5 x ULN. For the purposes of this study the ULN of ALT and AST is 45 U/L. * Total bilirubin ≤ ULN; or if \> ULN then direct bilirubin ≤ 1.5 x ULN * Female participants of childbearing age must have a negative pregnancy test at the time of enrollment. * Participants of childbearing or child fathering potential must be willing to use medically acceptable form of birth control during treatment and for 16 weeks after stopping treatment. * Participants and/or guardian have the ability to understand and the willingness to sign a written informed consent document according to institutional guidelines. Exclusion Criteria: Strata A and B * Participant who is receiving any other investigational agents. * Participants with other clinically significant medical disorders (i.e. serious infections or significant cardiac, pulmonary, hepatic, psychiatric, or other organ dysfunction) that could compromise their ability to tolerate protocol therapy or would interfere with the study procedures or results. * Participant with a history of clinically significant, uncontrolled heart disease and/or repolarization abnormalities as documented by a standard 12-lead ECG. * Shortening fraction of \<27% by ECHO or ejection fraction of \<50% by gated radionuclide study. * Prior history of QTc prolongation or QTc interval of \> 480 msec. * Female participants who are breastfeeding a child. * Participants are excluded if unable to comply with guidelines listed in appendix I.