Belmans, Oberdieck, and Rennemo asked whether all unnatural automorphisms of Hilbert schemes of points on surfaces, i.e. those automorphisms which do not arise from the underlying surface, can be characterized by the fact that they do not preserve the diagonal of non-reduced subschemes. Sasaki recently published examples, independently discovered by the author, of automorphisms on the Hilbert scheme of two points of certain abelian surfaces which preserve the diagonal but are nevertheless unnatural, giving a negative answer to the question.

We construct additional examples of unnatural automorphisms for abelian surfaces which preserve the diagonal for the Hilbert scheme of an arbitrary number of points. The underlying abelian surfaces in these examples have Picard rank at least 2, and hence are not generic.

We prove the converse statement that all automorphisms are natural on the Hilbert scheme of two points for a principally polarized abelian surface of Picard rank 1. Additionally, we prove the same if the polarization has self-intersection a perfect square