Action: Copy folder from private to public repo

Question

I want to create a GitHub Action for my private repository (repo A) that does the following:

• Check out a private repository (repo A)
• Check out a public repository (repo B)
• Cleanup subfolder bob of repo B
• Copy contents of subfolder repo A/alice to repo B/bob
• Push the change as a commit to repo B

Both are repositories from the same organization. I deployed a key pair (ACTIONS_SECRET) in the settings of my private repository like described in this answer. My current action looks like this:

name: Publish docsify

on:
  push:
    branches: [ main ]
  pull_request:
    branches: [ main ]

jobs:
  build:

    runs-on: ubuntu-latest

    steps:
    - name: Checkout alice
      uses: actions/checkout@v3
      with: 
        path: alice
    - name: Checkout bob
      uses: actions/checkout@v3
      with: 
        repository: <org/repo B>
        ssh-key: ${{ secrets.ACTIONS_SECRET }}
        ref: main 
        path: bob
    - name: Cleanup
      run: |
        pushd bob
        git rm -rf .
        popd
    - name: Copy alice
      run: |
        pushd bob
        cp -rvf ../alice/* . 
        popd
        ls -la bob/ 
    - name: Publish
      run: |
        eval `ssh-agent -s`
        ssh-add - <<< '${{ secrets.ACTIONS_SECRET }}'
        git config --global user.email "${GITHUB_ACTOR}@users.noreply.github.com"
        git config --global user.name "github-actions"
        pushd bob
        git add . 
        git commit -m "Build from Action ${GITHUB_SHA}" 
        git push origin main
        popd

The action fails at the Publish step with the following error:

ERROR: Permission to <repo B> denied to deploy key
fatal: Could not read from remote repository.

Please make sure you have the correct access rights
and the repository exists.
Error: Process completed with exit code 128.

This is my first action and I don't know how to solve this error. Please help.