PRs from forked repositories do not show results in "Files changed"

PRs originating from forked repositories do not show the results of the test in the "Files changed" tab.

Example:
https://github.com/AppImage/appimage.github.io/pull/2776

Hypothesis on what is going on:

When someone sends a PR from another (cloned) repository, then the script has no permissions to write the changes back to that other repository. Hence the changes can't appear in the PR.

In contrast, if I make a PR in this repo, then it works.

@andy5995 do you have an idea around this? (After all, I as the owner of the https://github.com/AppImage/appimage.github.io repo can make changes just fine if the cloned repo allows for it.)