A wife wants to attend religious classes. Does she need her husband's permission, and may he forbid her?

The essential knowledge needed for obligatory worship must be learned; if the husband does not teach it himself, he should not prevent his wife from learning it. For additional classes outside the home, good manners are to ask permission, and the husband is encouraged to give it.
The women asked the Prophet ﷺ to set aside a day for them because the men were taking all his time, and he gave them a day on which he taught them (Sahih al-Bukhari 101). Allah commands believers to protect themselves and their families from the Fire (at-Tahrim 66:6), which includes making sure a wife knows the basics of prayer, fasting, purification and the rulings of menstruation. A wife should choose classes that do not neglect household responsibilities and teachers who are recognised. Online classes are also a good option. See the question on a husband teaching his wife religion.