Missing userAttributes for federated Google User - amazon-web-services

Question

I'm trying to integrate Google as an IdP in our existing Cognito UserPool. Everything is set up so far, and I can SignUp/SignIn using Google, which creates the new user. I'm using the PreSignUp Lambda trigger to Link an existing user or create a new native one if there's no existing one. Now I was expecting that the event.Request.UserAttributes['name'] contains the user's name as provided by Google or at least seeing the attribute in the id_token. But I see no possibility to get those values at the moment. We started using Cognito just as the store for username/password, and none of the userAttributes are filled nor marked as required.
I have set up the Google integration with the following scopes:
.../auth/userinfo.email
.../auth/userinfo.profile
openid
In the UserPoolClient I:
marked name as read- and writeable attribute (along with others)
Checked the following allowed OAuth scopes email, openid, and profile. Those are also defined in the Web-Client in charge of the OAuth flow.
In the Federation section, I configured the attribute mapping:
Testwise, I mapped the name attribute to a custom attribute I used to test stuff. But neither this nor the mapping name to name worked.
Payload I get in the event:
{{PreSignUp_ExternalProvider .... Google_11...} {map[cognito:email_alias: cognito:phone_number_alias: email:m...#...m email_verified:true] map[] map[]} {false false false}}
id_token content:
{
"at_hash": "..",
"sub": "52...",
"email_verified": true,
"iss": "https://cognito-idp.us-west-2.amazonaws.com/...",
"cognito:username": "52..",
"origin_jti": "..",
"aud": "...",
"identities": [
{
"userId": "11...",
"providerName": "Google",
"providerType": "Google",
"issuer": null,
"primary": "false",
"dateCreated": "1648828708886"
}
],
"token_use": "id",
"auth_time": 1648828717,
"exp": 1648830828,
"iat": 1648830228,
"jti": "...",
"email": "m...#...m"
}
access_token content:
{
"origin_jti": "02...",
"sub": "52...",
"token_use": "access",
"scope": "openid profile",
"auth_time": 1648828717,
"iss": "https://cognito-idp.us-west-2.amazonaws.com/....",
"exp": 1648829317,
"iat": 1648828717,
"version": 2,
"jti": "..",
"client_id": "...",
"username": "52..."
}

Answer 1

Now it's working, even though I cannot state the error. I recreated the whole test set again and ensured the correct values of the following:
attribute mappings
authorized scopes
Allowed OAuth scopes
Scopes selected by the frontend
I added the scopes email, profile, and openid to be sure, and now I get the name attribute within the pre signup trigger lambda and in the ID-Token.