Configuring Apple Mail with Gmail Aliases
None of the other solutions worked for me right now with Apple Mail 9.3 on OS X 10.11.4. Just to make sure, my situation is as follows:


I have a primary email address xyz@somedomain.com as a Google Apps account
I added a domain alias otherdomain.com to that account
I want to send from xyz@otherdomain.com and have the receiver see this in the "from:" field (notice that this is slightly more complicated than in the original question, but includes the original question's case!)


After a lot of experimenting, this did work (both parts required, it seems).


On the Gmail end: 


For the primary email address xyz@somedomain.com, I went to Gmail online, then (under the top right tool icon) Settings > Account.
There I clicked: "Add another email address you own".
In the window popping up, I entered my (same) name and, under "Email address", the full alternative xyz@otherdomain.com
I left "Treat as an alias." unchecked.
Clicking the "Next step>>" button, I left the default top selection (the one that says "easier to set up")
I saved this.
(Back in the main Accounts screen, I checked "Reply from the same address the message was sent to", but that doesn't seem to be relevant to the root problem we are trying to solve here)



Using "Compose" still in Gmail online, I was able to change the from field to xyz@otherdomain.com and confirm that the receiver saw this as the "from:" field (as well as in the raw mail data).


Now in Mail, in the settings for xyz@somedomain.com I just added an alias in the "Alias:" field right under "Description:". Now Mail shows the alternative email address in the From field when composing emails in Mail, too, and the recipient sees xyz@otherdomain.com as the true sender (checked in the raw mail data again).


Notice that without the Gmail modification from Step 1, adding an alias to Mail just resulted in an additional line "X-Google-Original-From: xyz@otherdomain.com", which Mail wouldn't recognize as a sender in any form.