Auto-fork when when you don't have permissions to commit

I've started trying to submit PRs via Codespaces and the GitHub Pull Request functionality available through the extension.

There's a fundamental problem with this extension that the GitHub's web editor has already solved: if I don't have permissions to commit to a repository, I can't actually create a PR because I can't push a branch to the remote.

GitHub's web UI solved this by automatically forking and creating a branch for you (optionally allowing you to name the branch yourself, but defaulting to patch-n where n is an incrementing number based off of however many patch-n branches you already have, starting at 1).

Currently, to accomplish the same thing, I need to fork the repo myself (through the web or the gh CLI), change my origin, push to that, then make the PR from the pushed branch. That's a lot more work that is really unnecessarily painful.

Enabling users to have the ability to save their local changes to a fork and PR from that fork rapidly is essential to making Codespaces a good "quick PR" tool, and I'd really love to see it be enabled in that way that.