  Given an action of a finite group $G$ on the derived category of a smooth
projective variety $X$ we relate the fixed loci of the induced $G$-action on
moduli spaces of stable objects in $D^b(\mathrm{Coh}(X))$ with moduli spaces of
stable objects in the equivariant category $D^b(\mathrm{Coh}(X))_G$. As an
application we obtain a criterion for the equivariant category of a symplectic
action on the derived category of a symplectic surface to be equivalent to the
derived category of a surface. This generalizes the derived McKay
correspondence, and yields a general framework for describing fixed loci of
symplectic group actions on moduli spaces of stable objects on symplectic
surfaces.
