What happens to my bitcoins if there's a permanent fork in the chain?
The question is too speculative to answer. The groups could institute whatever rules they want.

If you assume there's a group that's willing to modify the software to create a permanent fork, presumably they are doing so for some reason. And whatever that reason is, that will affect what changes they make that cause the fork.

They could decide to allow people to transact on both chains. Or they could demand that you send the Bitcoins to a Bitcoin eater in the other chain before they allowed you to move them.