Just as the title suggests : If a user sends a message from Outlook and authenticates via SM the message should show in the sent items of webmail for that user.
Also, it would be nice since the message is going through SM if it could even pick up the signature settings. This would allow for branding regardless of the device used. I have used many signature managers for Exchage (CodeTwo etc.) that worked in this fashion. The sig just gets attached on the SMTP sink on the outbound connection.