The first step is to make it to that your sent messages appear on the mail server -- you can check in web mail, next closest thing.
( mail apps don't sync with each other, they all sync with your account data on the mail server )
And for that, you'll want to be sure that the account is set up in Outlook as IMAP or Exchange (not POP3, which doesn't provide access to server folders or the uploading of sent messages).