Is it permissible to inform a husband about his wife's misconduct that negatively affects his reputation and children, and is such informing considered a sin if it leads to divorce?

If the wife’s actions cause clear corruption and demonstrable harm to her husband, then you must first advise her and threaten to inform her husband or her family if she does not respond. If that does not work, inform someone who can restrain her, and informing the husband is permissible then, so you can be absolved of responsibility. However, if her actions do not exceed disclosing marital secrets that do not result in clear harm, and informing the husband might lead to her divorce, then you should only advise her without informing her husband.