Hosting a copy of on premise Exchange in Office 365 E3 plan

Hi guys

We have been trailing  the E3 plan and I have a quetion.

Would we be able to host a copy of our on premise exchange on office 365.

We would want to connect up our AD and then sync the user mailboxes up to office 365. Using it as say an edge server. 

Is that possible to have an exact copy of mailboxes online.

So the route would be mail is routed trough 365 and then down to on premise exchange. 

The reason for this would be when we redo a users PC for whatever reason the sync happens from the on premise machine and not from office 365.   some users have 20GB mailboxes and that would take a long time on our company's internet connection.

Thanks and Regards

Johan