There are no permissions you can set inside the container image to make this work. The workspace on the runner VM is mounted to /__w/ when the runner starts the job container (check the “Initialize containers” step in the log). Assuming a GitHub hosted runner VM the workspace is owned by the user runner. So what’s happening here is that the container is running as a different non-root user and can’t write another user’s files.

The documentation on Dockerfiles for Actions mentions the following regarding USER instructions:

Docker actions must be run by the default Docker user (root). Do not use the USER instruction in your Dockerfile, because you won’t be able to access the GITHUB_WORKSPACE.

If you use a job container you can’t adjust ownership before passing control to the container, so I’m afraid you’ll have to either run the container as root or start it manually (docker run and possibly docker exec).