Is it forbidden to claim that a certain woman is your mother while you know that she is not, and is it permissible to lie in such a situation to preserve the girl's honor and conceal the husband's sin?

If a woman breastfeeds a baby girl with a forbidden suckling (five sucklings before the age of two), then the wet nurse is considered the mother of the girl, and she is permitted to say "this is my daughter," intending the daughter through suckling.

If no suckling occurred, it is permissible for the woman to use equivocation (Tawriya) and say "this is my daughter," intending "like my daughter," or for the girl to say "this is my mother," intending respect and appreciation. There is no harm in calling her by that in front of people with the intention of concealing the matter and caring for the girl's feelings, and this is not considered a lie.

However, her explicit statement that she is her daughter by birth is a lie that is not permissible, and she must repent from it. But she is not required to deny it in front of people or correct what they have understood.