Does child custody transfer to the father if the custodial mother remarries? Does it return to her if the father remarries? Is it permissible for the divorced couple to agree upon the custody arrangement?

If the female custodian (mother) marries a non-mahram, her right to custody is forfeited, and custody transfers to the maternal grandmother according to the majority of scholars, or to the paternal grandmother according to some scholars, and to the father according to the Hanbalis. If the father marries, his custody is not forfeited. Scholars differed on whether custody is a right for the custodian or a duty upon them, and consequently, whether it is permissible to relinquish it. In cases of dispute concerning custody matters, the matter is referred to the Sharia court.