I set the registered usergroup so it can take quizzes, and some registered users can and others cannot. And they are in the same group, there are no differences between them. Do you know what could be the problem?

Not off the top of my head. Can you give me a temp admin account and let me know a username for someone who does work and someone who doesn't work?

---------- Post added at 23:31 ---------- Previous post was at 09:11 ----------

I see the same thing between the two users you gave me. Both of them can take quizzes. However you have their usergroup set so they it requires approval before they can take the quiz. So they click the take quiz button and it gives them a message that they must get approved before they can take the quiz. Maybe that's the problem?
if you want them to be able to take quizzes without having it to get it approved each time then for their usergroup set the Entry Auto Approve? setting to Yes.