Does a conditional divorce take effect if the husband falsely claims he told his wife: "If you go out to your family's house without my permission, you are divorced," when he never said that, and the wife did not go out without his permission?

If you falsely inform someone that you have made your wife's divorce conditional upon a certain action, then no divorce occurs even if the wife performs that action, because you did not swear an oath. Lying in this case necessitates sincere repentance, which involves abandoning the sin, regretting it, and resolving not to return to it.