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We study automorphisms of the Hilbert scheme of $n$ points on a genericprojective K3 surface $S$, for any $n \geq 2$. We show that the automorphismgroup of $S^{[n]}$ is either trivial or generated by a non-symplecticinvolution and we determine numerical and divisorial conditions which allow usto distinguish between the two cases. As an application of these results weprove 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 ageneralization of results by Boissi\`ere--Cattaneo--Nieper-Wisskirchen--Sartifor $n=2$.

Automorphisms of Hilbert schemes of points on a generic projective K3 surface