If you have another gmail acount (or you can create a new one) you can setup your main gmail acount to send copy of incoming mail to the secend acount.
Set your Q to download mail from the secend acount and send mail from your main acount.
This way you can trick gmail and avoid from this duplicate mails, and of course you will see all emails/response in your main acount when using the desktop computer.
I read this solution in an old post and its working great for me.
Good luck.