While I have occasionally given group quizzes, I usually use quizzes as a check of individual understanding when much of the other work for the course is occurring in a group setting. In courses where I have given more group quizzes, I have usually reduced the group testing, to maintain the balance between group and individual accountability. Students' reactions to group testing and group quizzes are usually very favorable. I occasionally use the strategy of a test that is taken individually but on which both an individual score and an average of group members' scores is given. However, I get some very useful information from doing this, especially in the C4L courses in which almost all work is done in groups.
Students find group tests less stressful.
I can't answer the above because there is no 'usual'. My group testing varies a lot.