  We study automorphisms of the Hilbert scheme of $n$ points on a generic
projective K3 surface $S$, for any $n \geq 2$. We show that the automorphism
group of $S^{[n]}$ is either trivial or generated by a non-symplectic
involution and we determine numerical and divisorial conditions which allow us
to distinguish between the two cases. As an application of these results we
prove that, for any $n \geq 2$, there exist infinite values for the degree of
$S$ such that $S^{[n]}$ admits a non-natural involution. This provides a
generalization of results by Boissi\`ere--Cattaneo--Nieper-Wisskirchen--Sarti
for $n=2$.
