In Outlook, when sending e-mail on behalf of someone else, it is not saved to the Sent Items of that user's mailbox. This is particular a problem when dealing with shared mailboxes, where both incoming and outgoing mail for those mailboxes need to be available for all users that require access to the mailbox.
If you are running Outlook with Exchange 2010 (since SP2 RU4) or up, in non-cached mode, do not use this method. You will trigger an issue with Outlook keeping sent items in the Outbox of the current mailbox. See MS KB 2703723 for more information.
To resolve this, depends on the Exchange version:
It's worth mentioning that Outlook with cached mode enabled does not suffer from this bug.
To resolve, create a registry value (manually, or through GPO):
Replace the "14.0" with the installed Office version (13 = 2007, 14 = 2010, 15 = 2013)
Restart Outlook for the changes to take effect (in case of GPO: gpupdate /force.