Hi @davido
Welcome to the Auth0 Community!
This kind of approach can be hard to tackle and implement within the Auth UI and I believe that Auth0 Lock will not be able to handle it out of the box unless you have a really hacky approach to it.
I would recommend to take a look at our Partials documentation regarding implementing some custom logic in order to replicate a HRD behaviour.
One other suggestion would be to create an OIDC connection for both Github and Google so that HRD can be configured on them and save you the hassle of coming up with the login when working with the Partials.
Otherwise, let me know if you have any other questions on the matter!
Kind Regards,
Nik