Not sure if already asked as I believe it's a common question:

Let's say that we have an existing customer service mailbox: support@example.com hosted somewhere. We have set up a support email address and forwarded all emails from support@example.com to this support email address, and also set up support@example.com as a "from address". Basically from a requester's perspective, they're only communicating with support@example.com

Now is there a way so that agent replies are copied to the "Sent" box (or whatever outbox) of support@example.com? Like how it would behave if you replied from Thunderbird IMAP-based mailbox? I have tried using the SMTP server of support@example.com but it seems that it only uses the SMTP server but does not copy the actual email to the outbox either.