The web server logs are probably the way to go.
The BEST advice is probably to find a way not to have to use quiz secure mode, however. That's never going to be entirely satisfactory. Unfortunately, that usually entails committing a large chunk of your life to writing hundreds of questions (you can always get the students to write them for you though!!).
If you get some useful data don't forget to let us know
Security and privacy
Identifying users browser?
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