She wants to raise an illegitimate child with her sons

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.