Flutter web Firebase Google authentication

I'm building a website for my project where I want to use Google auth by Firebase. I'm implemented a sign in method:

Future<User?> signInWithGoogle() async {
  User? user;
  GoogleAuthProvider authProvider = GoogleAuthProvider();
  try {
    final UserCredential userCredential =
        await auth.signInWithPopup(authProvider);

    user = userCredential.user;
  } catch (e) {
    print(e);
  }
  if (user != null) {
    name = user.displayName;
    imageUrl = user.photoURL;

    SharedPreferences prefs = await SharedPreferences.getInstance();
    prefs.setBool('auth', true);
  }
  return user;
}

My first question: Is this a good solution? I didn't find any good tutorial for this problem, just the Flutter documentation.Is it a safe solution for a website?

My second question: How can I sign out the user, and how can I detect if a user is signed in?