This article describes
When you invite someone that does not have an @vt.edu address to your SharePoint Site, they must accept the invite and authenticate from the same email account the invite was sent to or they will get an error.
One potential scenario that causes this:
They may have to forcibly log out of that Alpha@abc.com and make sure they attempt to authenticate with the proper one (in this example, Beta@abc.com).
There are some ways that the user can ensure they are accepting the invite and authenticating from the same account the invite was sent to:
Further, the owner of the site will not be able to see any information that the users have logged in correctly or incorrectly, they will only see the correct email that the original invite was sent to in the members list. If the user still cannot authenticate, the owner of the site can try to remove the user and add them again. Steps to remove a user from a SharePoint Site can be found at the How to delete users from a SharePoint site KB.