Center Controller reported confusing coordination with the TRACON for a military UAS approved to fly through a restricted airspace.

Narrative

Working ZZZ a handoff was initiated from ZZZ approach on Aircraft X climbing to the top of their stratum. Aircraft X was out of 080 for 100 with a requested final of FL200. I climbed Aircraft X to FL200. Aircraft X route of flight took him through Restricted Area X. I called ZZZ to confirm that his routing was approved through Restricted Area X as it was active up to FL220. The controller advised me that he was approved to transition through Restricted Area X at 100. I immediately hung up and descended Aircraft X back to 100. ZZZ called back and stated to keep him climbing if he was already through 110. Seeing the limited data block within the restricted area at 110 and knowing that Aircraft X was initiating a descent from 110; I coordinated a 20 degree left turn and climb to FL200. Following the call I issued the turn and climb to Aircraft X. After a incorrect read back I reissued the clearance and Aircraft X read it back and included traffic in sight. Coordination from ZZZ should have taken place; prior to transfer of control; with instruction on how Aircraft X would transition the restricted area or a heading to be flown to miss the restricted area.