If this woman is the mother of that girl it is not permissible for her - according to shareeah - to have custody of her Muslim daughter, because a kaafir cannot be entrusted with raising a Muslim child in Islam; rather he will cause the child to follow his own religion. We have discussed this previously in the answer to question no. 21516. So how about if she is not even her daughter?
It is shameful - let alone sinful - for a Muslim man to leave his children without any care and not bring them up in Islam, and to not care about anything except marriage without being prepared to take responsibility for its consequences. Fathers and mothers who neglect their children so that they end up becoming corrupt or apostatise from their religion are partners in this serious crime that affects their children, because they are the reason why it happens. Hence it is very important that you play a role through the local Islamic centre or some righteous people in the area, who could try to contact the father of that girl and inform him of his responsibilities towards his daughter, and advise him to take responsibility with his wife, and point out the dangers of neglecting this little girl.
We appreciate the questioner's keenness to do good and help people, and her concern about this Muslim girl's situation. It is clear that she is as upset as we are to see that the one who is taking charge of raising this Muslim girl is a Christian woman, who it is expected will turn her away from her natural state and true religion and cause her to become a Christian or to end up without any religion. Hence we advise the sister who asked this question to pay a great deal of attention to two important matters:
(i) After thanking this woman for coming to her and asking about the best way to raise a Muslim girl, she should try to get her to hand over custody of this Muslim girl to a Muslim family or an Islamic centre. Even better than that would be to look for Muslim relatives of this girl who are trustworthy in terms of their attitude and religious commitment, who could look after her and gave her a home. You have to be gentle and polite in doing this
(ii) If this woman does not agree to give up looking after this child, then we advise you to stay in close contact with her, to offer advice, care and education, so that this woman will not be the sole person concerned with this girl's upbringing lest she teaches her that which could lead to the corruption of her religious commitment or behaviour. For example, this woman could make arrangements for the girl to attend an Islamic school or spend time with a religiously committed family or with a woman who has good knowledge of Islam and works in dawah. If none of that is possible, then you should keep in touch with this girl so that you can communicate with her, so that she can remain in touch with her religion, her nature will not be spoiled and she will not apostatise from her religion. And seek reward for all of that with Allah, may He be exalted, and be patient in doing so, until Allah guides her parents - or one of them - to take her back and raise her in the true religion or the girl can go to a Muslim who can raise her such as a woman or an institution or a righteous person among her relatives.
Perhaps your being close to that woman will be a cause of her becoming Muslim, then she would be a suitable and trustworthy guardian to take care of that girl.
This is what we think you should do, and we encourage you to do it. We ask Allah to guide you and help you to do that which pleases Him.
See also How to raise a child in the answer to question no. 10016
And Allah knows best.
Reproduced from Islam QA