What is the religious ruling regarding a mother who refuses to custody her children and exposes them to ruin without a legitimate Shar'i excuse, while remaining unmarried?

Custody is a right of the custodian, and the mother may refuse it unless it is assigned to her. If it is assigned, she may not refuse. The mother should not be enabled to have custody if there is an impediment, such as her being an innovator or a transgressor. Refusing custody does not mean severing the mother's connection with her children.