I have a flutter library in a private GitHub repository I need to use as a dependency when running flutter pub get from a GitHub Actions workflow. How do I authenticate with git when doing this?

I have tried with dart pub token add like this:

- name: Get dependencies
  env:
    PUB_TOKEN: ${{ secrets.GITHUB_TOKEN }}
  run: |
    dart pub token add  --env-var "$PUB_TOKEN"
    flutter pub get

But I keep getting this error:

Resolving dependencies...
Git error. Command: `git clone --mirror  /home/runner/.pub-cache/_temp/dirKENEDH`
stdout: 
stderr: Cloning into bare repository '/home/runner/.pub-cache/_temp/dirKENEDH'...
fatal: could not read Username for '': No such device or address

exit code: 128
Failed to update packages.

How can I run the flutter pub get command while being authenticated with GitHub?