  G\"ottsche and Soergel gave formulas for the Hodge numbers of Hilbert schemes
of points on a smooth algebraic surface and the Hodge numbers of generalized
Kummer varieties. When a smooth projective surface $S$ admits an action by a
finite group $G$, we describe the action of $G$ on the Hodge pieces via point
counting. 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 resulting
generating functions give some interesting modular forms when $G$ acts
faithfully and symplectically on $S$.
