Is it permissible for a woman who was divorced three times by her husband, then married another man with the intention of returning to the first, and attempted to be divorced by the second but the divorce did not occur until after a period of time, to return to her first husband?

If a thrice-divorced woman marries with the intention of being divorced so she can return to her first husband, this is a matter of scholarly disagreement, with the preponderant opinion being that it is forbidden and that she does not become lawful for the first husband. However, if the second husband divorces you on his own accord, or at your request due to his ill-treatment, and not with the intention of returning to the first husband, then it is permissible for you to return to your first husband with a new contract after your waiting period (iddah) has ended.