All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
If the mother and the father have arrived at a certain arrangement as regards child custody then this is permissible. An example of this is that the child stays with each parent for a certain period. This is attested to in the following Hadith: Kathir bin 'Amr bin 'Awf Al-Muzani narrated from his father, from his grandfather, that the Messenger of Allah (PBUH) said: "Reconciliation is allowed among the Muslims, except for reconciliation that makes the lawful unlawful, or the unlawful lawful. And the Muslims will be held to their conditions, except the conditions that make the lawful unlawful, or the unlawful lawful." {Related by Tirmithi}.
Accordingly, reconciliation of mother and father must be in the best interest of the child.
However, if they failed to reach an agreement, then the mother has preferential right to custody until she gets married, in which case this right is transferred to the maternal grandmother, if she is capable of that. This is because it is conditioned that the person who takes custody is sane, devout, virtuous, and trustworthy as far as the interests of the child are concerned. If the latter wasn`t eligible, then custody is transferred to the maternal grand grandmother. If this wasn`t possible then custody is transferred to the paternal grandmother, then the paternal grand grandmother and so on. If this was also not possible then the father becomes entitled to custody of his child.
We (Iftaa` Department) recommend that you (Child parents) recourse to Sharia law by visiting one of the nearby Islamic centers or make reconciliation over this matter. And Allah The Almighty Knows Best.