If your primary Google account is main-email@domain.com and it has an assigned alias alias@domain.com , then it would be correct to do the following steps.
1) Create a new user in CRM with the address
2) Log out of the CRM. Log in to the CRM via Google Authentication as
main-email@domain.com .
3) Initiate integration with Gmail (sending section in settings)
4) Prescribe the alias.

After these steps you will be able to send mail from CRM using the alias.