You may have noticed that for objects copied natively (ADUC) the ‘Exchange Tasks’ menu in ARS is missing. But this behaviour is not present for objects copied in ARS.
This occurs by design. ARS will make the "Exchange Tasks" menu available based on the content of the msExchHomeServerName attribute.
The reason the copying behaviour differs from ARS and ADUC, is that ARS is designed to ignore the content of a couple of special attributes during the Copy User operation.
For objects copied in ADUC, please clear the content of the msExchHomeServerName attribute.