Tests are not automatically synced with the server. This is by design.
The app is aimed at students, some of whom may not have reliable internet access for many reasons. The use of a sync button means that students can control when and how the internet is accessed, for example, taking advantage of free wi-fi. It also means that tests can be downloaded and then done later, offline, when more convenient.
Similarly, teacher may want to make a large number of changes, for example importing set lists and adding new students. Having a sync button allows mistakes to be corrected safely before they are committed to the server where consistency requirements make things more difficult.
Students can download tests that they are set and information about the teaching groups they are in. Their answers are upload along with the calculations and working they did within the app along with the timing information.
Teachers have access to list of all students and teachers and, if they are given admin rights, can modify these lists. They can set and upload new tests for their teaching groups and download students answers as they become available. They can also see the tests and answers for other groups that they don't teach but which they are interested in keeping up to date with.