What is the ruling on a wife leaving to visit her family or relatives without her husband's knowledge and consent, knowing that her family approves of it?

It is not permissible for a woman to leave the house except with her husband's permission, whether it is from her marital home or her parents' home, and whether she is accompanied by her family or not. If your wife goes out to visit her sister accompanied by her family, and no corruption results from that, then there is no harm in permitting her this visit, and this does not diminish your authority. Understanding and respect should prevail between you, and you must strive to reconcile with your in-laws.