HI @adhadse, welcome
Authorizing a personal access token for use with SAML single sign-on is documented here:
HOWEVER this is only relevant if your Organization is configured for SSO to an IdP, which is not an available feature for user repositories or free plan Organizations. You will know if SAML is enabled for your organization as you have to provide IdP credentials in addition to your GitHub login credentials when accessing any organization repositories.
I am not overly familiar with actions, but the message you pasted states ‘The generated GITHUB_TOKEN (github_token) does not support to push to an external repository’. This implies the personal access token you created is not being used or configured correctly in your action code.
You can probably look around and find an example that you could follow to correct this problem, or another helpful community member may post a link.
I think you will need to specify the created personal access token secrets in the push to the external repository and not the default GITHUB_TON created by the workflow action, that does not have permission on external repositories