How to get Google OAuth 2.0 Client to pass client_id to AWS Cognito User Pool domain with Google Social Identity Provider

Question

I am getting an error when trying to sign-in to my single-page application (SPA) using the signInWithRedirect export of the aws-amplify/auth node package and using Google as a Social Identity Provider.

For my setup, I have done the following:

• created a set of OAuth 2.0 Client ID credentials on the Google Cloud > APIs and Services > Credentials page
• created a User Pool in AWS Cognito
• in my User Pool, under Social and external providers, I have added Google as an Identity Provider and I have added the Client ID and Client Secret from my Google OAuth 2.0 Client ID
• back in Google, for my OAuth 2.0 Client ID, I have added my User Pool's domain to the Authorized redirect URIs list
• in my Vue.js app, I have npm installed the aws-amplify@6.10.3 package
• in the source code of my app, I have the following (condensed) code:

import { Amplify } from 'aws-amplify'
import { signInWithRedirect } from 'aws-amplify/auth'

Amplify.configure({
  Auth: {
    Cognito: {
      loginWith: {
        oauth: {
          domain: '{MY_COGNITO_DOMAIN}',
          redirectSignIn: ['http://localhost:5173/'],
          responseType: 'code',
          scopes: ['email', 'openid']
        }
      },
      userPoolClientId: '{MY_USER_POOL_CLIENT_ID}',
      userPoolId: '{MY_USER_POOL_ID}'
    }  
  }
});

const signInWithGoogle = () => {
  signInWithRedirect({
    provider: 'Google'
  })
}

When the signInWithGoogle method is invoked in web application, the browser's network tab shows the following requests:

• a GET request to my Cognito User Pool's domain's /oauth2/authorize endpoint; this responds with a 302 HTTP status, redirecting to https://accounts.google.com/o/oauth2/v2/auth
• the GET request to the Google endpoint (above) responds with a 302, redirecting back to my Cognito User Pool domain, this time to the /oauth2/idpresponse endpoint
• the /oauth2/idpresponse Cognito endpoint responds with a 302, redirecting to a /error page on the same domain, which responds with a 400 - Bad Request

The Cognito error page that is the terminus of this flow displays the following message:

Missing required parameters
Missing required parameter client_id in request URL.

This is where I am stuck. As far as I understand, Cognito and Google do all the magic of constructing these redirect URLs and, presumably, each should be providing in the request parameters all of the data that the other requires. I don't know what I can change, or where, to get this client_id passed from Google to Cognito.