The movable cone of an irreducible holomorphic symplectic manifold admits a locally polyhedral wall-and-chamber decomposition which encodes information on the birational models of the manifold. In the case of moduli spaces of Bridgeland-stable objects on K3 surfaces, Bayer and Macrì provided a lattice-theoretical description of the walls in this decomposition, which allows for explicit computations. We will show how to apply these results to obtain a purely arithmetic classification of the automorphism group of Hilbert schemes of points on a generic projective K3 surface.