Counting polarizations on abelian varieties with group action

Let ${\mathcal{A}}_g$ be the moduli space of principally polarized abelian varieties. We study the problem of counting the number of principal polarizations modulo the natural action of the automorphism group of the abelian variety on a very general element of a positive dimensional component of $\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{g}({\mathcal{A}}_g)$, and show that this number is not always 1.