When I Reply to an Inbox message, does the Reply appear in the Sent items?

It seems that it doesn't.  I just replied to a student's message through InBox, but I do not see any record of that reply in the Sent items.

Is this intended behavior?

If so, is there any way to change this behavior? 

I would really like to view all of my sent email in the Sent items, if you see what I mean.

Thanks.