To add an email alias, please see this article Adding email alias in Google Apps
If you are getting this error when sending email to the email alias address:
Delivery to the following recipient failed permanently:
--- email alias address---
Technical details of permanent failure:
The partner did not specify the domain to relay the emails
Please check if your users have the correct email routing enabled. Do the following:
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Go to admin.google.com
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Select 'Users' and on the next page select the account with the email alias you had attempted to send the email to.
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Select 'Account' and scroll down to 'Email routing'.
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Ensure that 'Google Apps email' is selected as a destination (see screenshot below)
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If you are having issues enabling Google Apps email routing , please contact support.