How do I use authetnication from site in my game?

I'm making a promo game that will be hosted on a website. The game will be available to everyone, but if you're already logged in on the site (you can be logged in through facebook account, site account, google etc)

I don't want to have another login inside of the game with credentials. Is there any way to get the user data from the site you're already logged in on, and how?

Any help appreciated.