Once the students have added items, they can be previewed and edited by the teacher & other group members. In a group, members can view what an item looks like, but actual interaction for something like a quiz happens in a course. This lets the teacher decide what items to bring into a course for student interaction.
So what can students do as admins? In our system, we have it set so that they can add things like updates and resources, but they cannot (even as admins) change group settings, message anyone other than teachers, and delete the group. While they can delete items in a group (like links or files), anything deleted goes into the Recycle Bin for the Group which can be restored by the teacher.
While Schoology is one way for students to generate questions, there are many other options to accomplish this, including Google Forms. If you need help or have questions about how you might do this with students, please contact your school's technology coordinator or your Technology & Learning Coach.