page redirect to external site with authentication

I have a WP site where there are only authorized users, with a plug in showing their personal energy data based on user ID.

I want to offer these users a link to another external site which is password protected and the URL is personalized to the user ID.

How can I create a blank page or something within WP that will automatically redirect to the external site while carrying through their login credentials (which I have).

Example for clarity.

user1 is signed in my WP with ID=user1 and pwd1.
user1 clicks on generic WP page [external] which redirects user1 to https://www.abcuser1.example.com while carrying through the authentication which is also user1/pwd1 on the external site.

user2 is signed in my WP with ID=user2 and pwd2.
user2 clicks on generic WP page [external] which redirects user2 to https://www.abcuser2.example.com while carrying through the authentication which is also user2/pwd2 on the external site.

The server on the external site is apache and understands curl requests.

Thanks !!