Most decently large colleges have been investing in HPC for a while, and started investing in GPU HPC around 2014. You'd be surprised what sort of school projects the compute budget exists for.

I went to a smallish state university, even there we had our own HPC center and lab. We had a proper HPC (IIRC) 6 row data center across campus and we had a continuous budget available to me as an undergraduate research assistant for building beowulf clusters for the graduate programs to run assignments on. I once got an allowance to buy 15 raspberry pis to build an arm cluster.

As far as research groups go - they get funds (project grants, donations, etc.) to purchase machines and parts, and then users have to timeshare them.

These machines are pretty much crunching numbers 24/7, and your project will get appended to a queue.

'group project'