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We determine the possible finite groups $G$ of symplectic automorphisms ofhyperk\"ahler manifolds which are deformation equivalent to the second Hilbertscheme of a K3 surface. We prove that $G$ has such an action if, and only if,it is isomorphic to a subgroup of either the Mathieu group $M_{23}$ having atleast four orbits in its natural permutation representation on $24$ elements,or one of two groups $3^{1+4}{:}2.2^2$ and $3^4{:}A_6$ associated to$\mathcal{S}$-lattices in the Leech lattice. We describe in detail those $G$which are maximal with respect to these properties, and (in most cases) wedetermine all deformation equivalence classes of such group actions. We alsocompare our results with the predictions of Mathieu Moonshine.

Finite groups of symplectic automorphisms of hyperk\"ahler manifolds of type $K3^{[2]}$