Praise be to Allah.
This child is illegitimate, i.e., the child of zina (adultery/fornication), and as such comes under the rulings on foundlings (laqeet), so long as his or her mother has died. It is a form of kindness to this child to keep him and look after him. If the Muslim sister takes this child in with her own children and brings him up, this is an act of kindness and a righteous deed, but he will not be counted as one of her children or one of her husband’s children, rather he will be a stranger (ajnabi), unless she breastfeeds him, giving him five complete feedings, whilst he is still an infant; in that case he will become a son to her through radaa’ah (breastfeeding) and a son to her husband through radaa’ah. Her sons will also be brothers to him through radaa’ah, but there will be no rights of inheritance between them, because the relationship between them will be one of having been raised together and of kindness, or of brotherhood through radaa’ah, none of which bestows rights of inheritance between them and this child.