Cannot update submodule on self-hosted runner using SSH

I have a self-hosted runner on windows server. It is running as a default service, username: Network Service. While configuring the runner, I selected all default options.

I have a repository with one submodule in it.
The runner fails while updating the submodule in the main repository.

runs-on: self-hosted
steps:
  - name: Check out repository
    uses: actions/checkout@v2
    with:
      fetch-depth: 0
      submodules: true
      ssh-key: ${{ secrets.KEY }}

Start with the workflow using actions/checkout@v2 on the Windows self-hosted runner configured as Network Service, and reproduce the failure with submodules: true and ssh-key. Compare the runner's SSH and submodule behavior with a successful checkout; done means the submodule updates over SSH in this configuration.