Is it possible to share auth between Chrome extension and website?

I have a Chrome extension that uses Firebase authentication (via firebaseui), and a corresponding hosted site that also uses Firebase for authentication (also via firebaseui). How would I go about making the user automatically sign in on the website, if they are already signed in in the Chrome extension?

The extension's authentication lives in its background page, and I can post message from that to the content script, which in turn can post messages to the website (or the other way around), but what do I need to pass over to have the site (which I can control) have enough information to directly sign the user in?