  In this paper we study equivariant moduli spaces of sheaves on a $ K3 $
surface $ X $ under a symplectic action of a finite group. We prove that under
some mild conditions, equivariant moduli spaces of sheaves on $ X $ are
irreducible symplectic manifolds deformation equivalent to Hilbert schemes of
points on $ X $ via a connection between Gieseker and Bridgeland moduli spaces,
as well as the derived McKay correspondence.
