This is pretty much the same as Reply to email from forwarded email account using original email address except, apparently, that I'm using Gmail as the forwarding target. I've done the settings suggested by https://superuser.com/a/733268/423386. Neither worked for me.
I'm using Outlook for MS Office 365. For a personal email account I use a forwarder so my address is firstname.lastname@example.org. This is forwarded to a Gmail account, email@example.com, that has IMAP/SMTP access enabled. I'd like people I correspond with to only use the forwarder address since the account it's fowarding to can change. So when I send an email, it should show the sender as firstname.lastname@example.org.
I've set up the Outlook account for the email@example.com, with "Your name" set to "me" and "Email address" set to "firstname.lastname@example.org". However, when I send an email, the sender is shown as "email@example.com". How can I fix this?
I'm not trying to fool anyone here. Someone who cares can easily find out the Gmail address. What I want is that a recipient can just click Reply and the reply will be sent to the forwarder address, not the Gmail address.
I've done this very thing successfully when forwarding to an email account, firstname.lastname@example.org, on a hosting service and it works fine. But it doesn't work for Gmail. Is there something special about Gmail that's causing this? Or am I missing something?