What is the ruling of Islamic law concerning a woman who adopted a foundling and attributed him to herself to avoid shame and to preserve her life and the lives of her family from her tribe, and what is the legal way out for her from this sin while preserving her modesty and the reputation of her daughters?

A person of unknown lineage is considered an orphan, and sponsoring them earns the same reward as sponsoring an orphan, due to the generality of the Hadith: "I and the sponsor of an orphan will be in Paradise like this," (and he gestured with his index and middle fingers).

There are two scenarios: 1. If the child (under two years old) is breastfed five full feedings: He becomes your son through rada'a (wet-nursing), your husband becomes his father, and your children become his siblings through rada'a. He is permitted to mix with you. However, his lineage cannot be attributed to your husband, as this contradicts Allah's saying: "Call them by [the names of] their fathers; it is more just in the sight of Allah." (Quran 33:5). A general name should be chosen for him, such as so-and-so bin Abdullah. 2. If he is not breastfed, or breastfed fewer than five times, or is over two years old: He remains a non-mahram to you. The mother and her daughters must observe hijab in his presence when he reaches puberty or approaches it.

The child's lineage must be changed if it has been attributed to your husband, unless there is a confirmed harm in doing so, in which case the change can be delayed until a suitable opportunity arises.