The equivariant Brauer groups of commuting free and proper actions are   isomorphic
  If $X$ is a locally compact space which admits commuting free and proper actions of locally compact groups $G$ and $H$, then the Brauer groups $\Br_H(G/X)$ and $\Br_G(X/H)$ are naturally isomorphic. 
Alexander Kumjian, Iain Raeburn, and Dana P. Williams