How to make authorization through the site in the browser extension?

Question

Google Chrome

I have an extension that checks the site's URL in the database, and if there is a match, it displays information about it.
I also made a conclusion to the site page of a small div block with a link through which the client goes to my site and logs in. Then it is transferred back, while receiving a referral ID specifically for this site.
Here is the question. How can I simultaneously transfer data to the extension during this authorization on the site? No need to log in twice. Immediately take the information from the extension.
Extension example: LetyShops.
Didn't find any info on this

Answer 1

the question is unclear
but look at some semrush | ahrefs