Flutter Web: Login status and router settings in Firebase Multisite

Flutter Web: firebase google login

Login status and router settings in Firebase Multisite. login.abc.com and account.abc.com use the same resource. When you log in to login.abc.com, you are taken to account.abc.com, and when you log out of account.abc.com, you are taken to login.abc.com and the status is switched to Logout. How do I implement it in flutter web?

ex:> I want to implement it like https://www.squarespace.com/.