Is expiation obligatory and the oath dissolved, knowing that the wife did not go to her friend's house so that she could go whenever she wanted with her husband's permission?

Likening one's wife to one's mother in terms of prohibition is a reprehensible and forbidden statement, and it is considered a conditional ẓihār. If the wife goes to her friend's house, the ẓihār takes effect, and it becomes unlawful to have intercourse with her until the expiation for ẓihār, as mentioned in the Qur’an, is performed: freeing a slave; if one cannot find one, then fasting for two consecutive months; if one is unable to do that, then feeding sixty poor people. The expiation is not valid before the cause of the ẓihār (the wife’s departure) occurs.