Firebase JWT Authentication - Adding user information

I am using AWS Cognito, with a lambda function to create a custom JWT so that I can connect to my Firebase database.

It works, it creates a new user in Firebase Auth, but only with its UID.

I would like to add information to this token, so that:

• In Firebase Auth dashboard, the email of the user is displayed

• In FlutterFlow, I can access these values via the "Authenticated User" variable

Any idea to do it?

Find attached my lambda function, as well as my function to decode the JWT before login to Firebase with it.