Emails sent through the webmail interface are not showing up in my Sent Items folder?

This happens when you do not have the 'Save copy of replies in Sent Items folder' option enabled on your webmail account.

To enable this option log into your webmail account click on the Settings menu option -> My Settings. Next select the Compose tab, then tick the checkbox for the option 'Save copy of replies in Sent Items folder'. Finally click save.