If your primary Google account is firstname.lastname@example.org and it has an assigned alias email@example.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
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.