Firebase Get Existing User Google

I want to check if the user (when google signing in) already exists in firebase. In my app, when the user signs up, I call signinwithgoogle, which runs this method in my code

Future<User?> signInWithGoogle() async {
  await Firebase.initializeApp();

  User? user;

  if (kIsWeb) {
    GoogleAuthProvider authProvider = GoogleAuthProvider();

    try {
      final UserCredential userCredential =
          await _auth.signInWithPopup(authProvider);

      user = userCredential.user;
    } catch (e) {
      print(e);
    }
  } else {
    try {
      final GoogleSignIn googleSignIn = GoogleSignIn();

      final GoogleSignInAccount? googleSignInAccount =
          await googleSignIn.signIn().catchError((onError) => print(onError));
      ;

      if (googleSignInAccount != null) {
        final GoogleSignInAuthentication googleSignInAuthentication =
            await googleSignInAccount.authentication;

        final AuthCredential credential = GoogleAuthProvider.credential(
          accessToken: googleSignInAuthentication.accessToken,
          idToken: googleSignInAuthentication.idToken,
        );

        try {
          final UserCredential userCredential =
              await _auth.signInWithCredential(credential);

          user = userCredential.user;
        } on FirebaseAuthException catch (e) {
          if (e.code == 'account-exists-with-different-credential') {
            print('The account already exists with a different credential.');
          } else if (e.code == 'invalid-credential') {
            print('Error occurred while accessing credentials. Try again.');
          }
        } catch (e) {
          print(e);
        }
      }
    } catch (e) {
      print(e);
    }
  }

  if (user != null) {
    uid = user.uid;
    name = user.displayName;
    userEmail = user.email;
    imageUrl = user.photoURL;

    SharedPreferences prefs = await SharedPreferences.getInstance();
    prefs.setBool('auth', true);
    prefs.setString('id', user.uid);
    prefs.setString('name', user.displayName!);
  }

  return user;
}

HOWEVER, this doesn't tell me if the user already exists. When I call this method, I want it to tell me if the user is new, or just went through the entire sign up process by accident. If the user went through the sign up process by accident, when signing up with google he should automatically be redirected to the home page. How would I check that?

You can use this method:

   Future<bool> checkExistUser(String email) async {
    final response = await FirebaseAuth.instance.fetchSignInMethodsForEmail(email);
    return response.isNotEmpty;
  }

this code works for me.