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For an Abelian surface $A$ with a symplectic action by a finite group $G$,one can define the partition function for $G$-invariant Hilbert schemes \[Z_{A,G}(q) = \sum_{d=0}^{\infty} e(\text{Hilb}^{d}(A)^{G})q^{d}.\] We prove thereciprocal $Z_{A,G}^{-1}$ is a modular form of weight $\frac{1}{2}e(A/G)$ forthe congruence subgroup $\Gamma_{0}(|G|)$, and give explicit expressions interms of eta products. Refined formulas for the $\chi_{y}$-genera of$\text{Hilb}(A)^{G}$ are also given. For the group generated by the standardinvolution $\tau : A \to A$, our formulas arise from the enumerative geometryof the orbifold Kummer surface $[A/\tau]$. We prove that a virtual count ofcurves in the stack is governed by $\chi_{y}(\text{Hilb}(A)^{\tau})$. Moreover,the coefficients of $Z_{A, \tau}$ are true (weighted) counts of rationalcurves, consistent with hyperelliptic counts of Bryan, Oberdieck,Pandharipande, and Yin.

$G$-invariant Hilbert Schemes on Abelian Surfaces and Enumerative Geometry of the Orbifold Kummer Surface