I understand that you want to implement a Single Sign On using Cognito for external Users programatically.
Programatically this can be implemented by any Framework of your choice. You can use the Amplify Framework using the attached document.
As a workaround, I have replicated the above statement using Angular Framework.
- Create and configure a user pool on Amazon Cognito.
- Create an Angular application.
- Add the signin and signup pages in the Angular application.
- Add the Angular application and validated the user's signup on Amazon Cognito.
I have attached a third party documentation  to programatically implement the above mentioned, you can also implement the above by React Framework.
I hope this helps
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