Is it permissible for a Muslim woman who has children and a husband to raise an orphaned child whose mother is Muslim and father is not Muslim, and who has no Muslim relatives?

If the mother of a child born out of wedlock dies, the child's ruling is that of a foundling. Custody and care for such a child are considered acts of kindness and good deeds. A Muslim sister can take this child in to live with her own children, providing care and upbringing. However, the child remains a stranger to her and her husband unless she breastfeeds the child five times during the nursing period. In that case, the child becomes her and her husband's son through fosterage, and a foster-sibling to her children. There is no inheritance between them, whether through upbringing and kindness or through fosterage.