  G\"ottsche gave a formula for the dimension of the cohomology of Hilbert
schemes of points on a smooth projective surface $S$. When $S$ admits an action
by a finite group $G$, we describe the action of $G$ on the Hodge structure. In
the case that $S$ is a K3 surface, each element of $G$ gives a trace on
$\sum_{n=0}^{\infty}\sum_{i=0}^{\infty}(-1)^{i}H^{i}(S^{[n]},\mathbb{C})q^{n}$.
When $G$ acts faithfully and symplectically on $S$, the resulting generating
function is of the form $q/f(q)$, where $f(q)$ is a cusp form. We relate the
Hodge structure of Hilbert schemes of points to the Hodge structure of the
compactified Jacobian of the tautological family of curves over an integral
linear system on a K3 surface as $G$-representations. Finally, we give a
sufficient condition for a $G$-orbit of curves with nodal singularities not to
contribute to the representation.
