Very symmetric hyper-Kähler fourfolds

G. Höhn and G. Mason classified all finite groups acting faithfully and symplectically on a hyper-K{ä}hler fourfolds of type K3${}^{[2]}$. There are 15 maximal among them, call them ${\tilde{G}}_1,\dots ,{\tilde{G}}_{15}$. Every manifold of type K3${}^{[2]}$ admitting an action of ${\tilde{G}}_i$ for some $i$ must necessarily have Picard rank 21 which is maximal. This fact allows us to use lattice-theoretic methods to classify all the finite groups $G$ acting faithfully on a hyper-K{ä}hler fourfold of type K3${}^{[2]}$ $X$ such that $G$ contains ${\tilde{G}}_i$ as a proper subgroup and ${\tilde{G}}_i$ acts symplectically on $X$. We also describe examples of fourfolds of K3${}^{[2]}$-type admitting an action of such groups.