Amplify Auth user pool custom domain doesn't get reflected in the generated amplify_outputs.json

[amalhub]

Environment information

System:
  OS: Windows 10 10.0.19045
  CPU: (16) x64 Intel(R) Core(TM) i7-10875H CPU @ 2.30GHz
  Memory: 13.03 GB / 31.77 GB
Binaries:
  Node: 20.15.1 - C:\Program Files\nodejs\node.EXE       
  Yarn: undefined - undefined
  npm: 10.3.0 - C:\Program Files\nodejs\npm.CMD
  pnpm: 9.5.0 - ~\AppData\Roaming\npm\pnpm.CMD
NPM Packages:
  @aws-amplify/auth-construct: 1.3.0
  @aws-amplify/backend: 1.2.0
  @aws-amplify/backend-auth: 1.1.3
  @aws-amplify/backend-cli: 1.2.5
  @aws-amplify/backend-data: 1.1.3
  @aws-amplify/backend-deployer: 1.1.0
  @aws-amplify/backend-function: 1.3.4
  @aws-amplify/backend-output-schemas: 1.2.0
  @aws-amplify/backend-output-storage: 1.1.1
  @aws-amplify/backend-secret: 1.1.0
  @aws-amplify/backend-storage: 1.1.2
  @aws-amplify/cli-core: 1.1.2
  @aws-amplify/client-config: 1.3.0
  @aws-amplify/deployed-backend-client: 1.4.0
  @aws-amplify/form-generator: 1.0.1
  @aws-amplify/model-generator: 1.0.5
  @aws-amplify/platform-core: 1.0.7
  @aws-amplify/plugin-types: 1.2.1
  @aws-amplify/sandbox: 1.2.0
  @aws-amplify/schema-generator: 1.2.1
  aws-amplify: 6.5.4
  aws-cdk: 2.155.0
  aws-cdk-lib: 2.155.0
  typescript: 5.5.4
AWS environment variables:
  AWS_NODEJS_CONNECTION_REUSE_ENABLED = 1
  AWS_SDK_LOAD_CONFIG = 1
  AWS_STS_REGIONAL_ENDPOINTS = regional
No CDK environment variables

Describe the bug

I have an Amplify Gen 2 NextJS app configured with an external Google OAuth provider. After purchasing a custom domain for my website externally, I successfully configured the app with the custom domain so that when I go to my custom domain from my browser the website homepage loads successfully.

Now, when I log in using Google OAuth, I am redirected to the OAuth consent screen, but it still shows the redirect URL as <userpool>.auth.<region>.amazoncognito.com. To change this I followed the below steps:

Steps:
Set Up Your Custom Domain in AWS Cognito:

Go to the Cognito Console and select your User Pool.
Under the App Integration section, choose Domain name.
Click Use your own domain and enter your custom domain (e.g., auth.yourdomain.com).
Set Up the DNS Record:

In your DNS management console (e.g., Route 53), create a CNAME record.
Point your custom domain (e.g., auth.yourdomain.com) to the AWS Cognito service domain (e.g., your-userpool-id.auth.region.amazoncognito.com).
Verify Your Domain:

Once the DNS changes propagate, Cognito will verify the domain. This can take up to 24 hours.
Update Your Google OAuth Configuration:

Log in to the Google Cloud Console.
Go to APIs & Services > Credentials.
Edit your OAuth 2.0 Client ID and update the Authorized redirect URIs to use your custom domain. 

Still it doesn't change the redirect URL because the auto generated amplify_outputs.json still has the amazon default domain <userpool>.auth.<region>.amazoncognito.com. I even tried redeploying the app, still it doesn't pick up the custom domain I configured in the userpool.

I there a way to configure this in the Amplify Gen 2 NextJS backend auth configuration? Or is this a bug?

Reproduction steps

Steps to reproduce given above.