Is it permissible for a divorced woman to hand over her children to her ex-husband for the purpose of marriage, even if the father refuses, and there is the paternal grandmother (the husband's mother) and a private home with her?

The default rule for child custody is that it belongs to the mother as long as she does not remarry. If she remarries, the right of custody transfers to the child's closest relatives. The mother has the primary right to custody, and it is both her right and obligation if the child needs her and there is no one else. If the mother relinquishes custody of her children for the sake of marriage, the father is then obligated to fulfill his responsibility towards his children. The reference for matters of custody is the legitimate ruler (Islamic judge), who considers the best interest of the child under custody.