That's precisely why the cookie should just be an identifier, that you look up group info from the database. Because you can guarantee the cookie contents will be modified by someone at some point. Make it useful to you, useless to them.
Or if you can bruteforce the secret, or if there's a vulnerability in the secret, or if... You're relying on the fact that the cryptography will be impregnable, rather than adopting an actual security posture.
Do not trust the data you send to a user, to remain secure.
No. It's relying on both cryptography, and the inaccessiblity of information. Which is a tried, practiced, and often federally mandated, method of security. Controlling who has access to information is sorta security 101. Don't dump your database to the Internet.
Security through obscurity is allowing REST commands to the /totallysecretaddress/neverleaked/ URI.