Is there sin upon one who informs her husband, who is one of her mahram relatives, about his wife's disobedience to him and her leaving the house without his permission, and her doing what he forbade her from in his absence, after repeated advice proved futile?

If you advise a woman and she does not respond, threaten her with informing her husband about what she is doing. If she still does not desist, you are permitted to inform him, provided that no greater harm results from this than her going out without his permission. If informing him is the only way to prevent her from committing evils, then it is obligatory for you to inform him, based on the saying of the Prophet, peace and blessings be upon him: "Whoever among you sees an evil, let him change it with his hand; if he cannot, then with his tongue; if he cannot, then with his heart." However, if informing her husband would lead to a greater harm, then it is not permissible for you to do so, and you must continue to advise her.