I am not too  familiar with how outlook works but could it be that you have more than one email addresses set up in Outlook and therefore there are more than one 'Sent Item' folders.

Another way to check this is to type in the invoice nbr in the search box and make sure to also click on 'Try searching again in All Mail Items' link at end of column.

I am running the same setup as you have listed and have no problem seeing it in my sent box. I did a test and sent an invoice to myself and it shows up in both the inbox and sent mail box.

It's possible to have different settings for different email accounts in Outlook.  

If all other sent mails are showing in your sent mail, and only Sage 50 emails aren't, they may be going through a separate email account.

If no sent mail is showing, the Outlook setting may be to either not keep a copy of sent email, to not keep a copy in the local email post office, or to empty the sent mail folder on closing Outlook, or to only keep a copy on the server.

I am having the exact same issue.  Running Win 7 Pro 64 bit, Sage 50 2015, Outlook 2013 32 bit.  If I email the invoice to myself then forward to the client the email is in my sent mail folder - this is the only way I can confirm that the email has been sent.   If I try to email directly from Sage to the client, the email does not show up in my sent mail folder.  A fix would be nice.  

Thanks Smithco:  After reading your suggestion to search other sent folders.  I found all my invoice emails in the Archive "Sent Items" Folder.  

I am now trying to figure out where you change the default "Sent Items" folder from the Archive account to my email account.  Any help would be appreciated.