Display actions in fork in the pull request

Question

Long story short, I needed to use pull_request_target because I needed to checkout a private repo and you can't transfer secrets to a fork.

So now, in my .github/workflow.yml file I have the following lines:

on: [push,pull_request_target]

jobs:
  build-and-test:
    runs-on: ubuntu-latest

    steps:
      # Checkout the code of the current branch
      - name: Checkout code
        uses: actions/checkout@v2

      - name: Checkout private repo
        uses: actions/checkout@v2
        with:
          repository: private/repo
          path: './private'
          token: ${{ secrets.MY_PAT }}

The thing is, this will run on my fork's branch, not on the pull request so I can't see directly in the pull request screen if everything is ok.

This is a commit before using pull_request_target, notice the red cross because some checks failed. The second one is a commit after using pull_request_target, I don't even have a cross or check, I need to click on the commit then on my branch then in the actions tab.

Is there a way to push the results from my branch to the pull request on the main repository?