Let's say I have a submodule which is cloned from a Github repo where I can't push to. I make changes to this submodule and commit them locally.
Now, I push my repo (which contains the submodule) to my Github account so others can clone from it. But I also want others to see/use my commits I made to the submodule. When I push my repo, the commits to the submodule are somehow not pushed (I wonder how this should work, anyway). Whenever somebody clones from my repo, they get the last commit of the submodule's authors, not my commits.
What do I have to do in order to make this work?
No, this will not work because you cannot create git repository inside one git repository. This means you cannot publish two different repositories commits into another git repository.