  We study the equivariant sheaf counting theory on K3 surfaces with finite
group actions. Let $\sS=[S/G]$ be a global quotient stack, where $S$ is a K3
surface and $G$ is a finite group acting as symplectic homomorphisms on $S$. We
show that the Joyce invariants counting Gieseker semistable sheaves on $\sS$
are independent on the Bridgeland stability conditions. As an application we
prove the multiple cover formula of Y. Toda for the counting invariants for
semistable sheaves on local K3 surfaces with a symplectic finite group action.
