Is it permissible for a woman to falsely swear by Allah before her husband, who has divorced her, if he asks her whether the woman her husband cheated with had told her the details of the betrayal, in order to cover for this woman and protect her home?

It is obligatory for a woman to conceal herself and not disclose any disobedience committed by her or her husband. The presumption for your husband is innocence until proven otherwise by clear evidence. You have erred by believing the woman without verification, for Allah Almighty says: "O you who have believed, if there comes to you a disobedient one with information, investigate, lest you harm a people out of ignorance and become, over what you have done, regretful." Lying is forbidden, but if it becomes the only means to achieve a legitimate benefit or avert harm, then it is permissible.