Are the wife's family entitled to keep their daughter's gold, which the husband returned to them, especially since the wife's father is an Imam of a mosque? Is this act permissible or forbidden, given that there is a dispute between the spouses because of it?

The gold is the wife's property, and she has the right to demand it from her family or to absolve them of it. If the wife lends the gold to her husband for him to return it to her, and he gives it to her family, then he has the right to demand it back. Such disputes are resolved by consulting scholars directly or by referring them to the judiciary.