  Belmans, Oberdieck, and Rennemo asked whether natural automorphisms of
Hilbert schemes of points on surfaces can be characterized by the fact that
they preserve the exceptional divisor 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 exceptional divisor but are nevertheless unnatural, giving a
negative answer to the question. We construct additional examples for abelian
surfaces of unnatural automorphisms which preserve the exceptional divisor of
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.
