given an automorphism of a variety $x$, there is an induced ''natural'' automorphism on $x^{[n]}$, the hilbert scheme of $n$ points of $x$. while unnatural automorphisms of $x^{[n]}$ are known to exist for certain varieties $x$ and integers $n$, all previously known examples could be shown to be unnatural because they do not preserve multiplicities. belmans, oberdieck, and rennemo thus asked if an automorphism of a hilbert scheme of points of a surface is natural if and only if it preserves the diagonal of non-reduced subschemes.

we give an answer in the negative for all $n\ge 2$ by constructing explicit counterexamples on certain abelian surfaces $x$. these surfaces are not generic, and hence we prove a partial converse statement that all automorphisms of the hilbert scheme of two points on a very general abelian surface are natural for certain polarization types (including the principally polarized case).