What is the legal ruling for agreeing with another man to marry the mother with the intention of divorcing her the next day so that she can return to her first husband? And what is the ruling regarding the mother remaining with her current husband after this incident? And in the event that it is impermissible for them to remain together, what is the expiation or the obligatory act upon them?

If a man divorces his wife for the third time, she is not lawful for him until she marries another husband in a valid marriage of desire, and then he divorces her or dies. This is based on the Almighty's saying: (And if he has divorced her [for the third time], then she is not lawful to him afterward until she marries another husband). As for "tahlil" marriage, which is marriage with the intention of making the woman lawful for her first husband, it is forbidden and void by scholarly consensus, and the woman does not become lawful for her first husband through it. This is based on the Prophet's (peace be upon him) saying: (May Allah curse the one who performs tahlil [makes something lawful] and the one for whom tahlil is performed). Whether the second husband openly declares his intention or merely intends it in his heart, the contract is void, and the woman does not become lawful for her first husband through it. Moreover, merely the marriage contract does not make her lawful for the first husband until each of them tastes the sweetness of the other (i.e., sexual intercourse). This is confirmed by the hadith of Aisha and the statements of Al-Nawawi and Ibn Qudamah. Based on this, if your mother's marriage was for the purpose of tahlil, it is legally void, and she must separate from this husband.