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G\"ottsche and Soergel gave formulas for the Hodge numbers of Hilbert schemesof points on a smooth algebraic surface and the Hodge numbers of generalizedKummer varieties. When a smooth projective surface $S$ admits an action by afinite group $G$, we describe the action of $G$ on the Hodge pieces via pointcounting. Each element of $G$ gives a trace on$\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$.In the case that $S$ is a K3 surface or an abelian surface, the resultinggenerating functions give some interesting modular forms when $G$ actsfaithfully and symplectically on $S$.

Hilbert schemes of points on smooth projective surfaces and generalized Kummer varieties with finite group actions