Very symmetric Hyperkahler fourfolds

G. H ̈ohn and G. Mason classified all finite groups acting faithfully and symplectically on a hyper-K ̈ahler fourfold of type K3[2]. Fifteen of them are maximal, we call them eG1, . . . , eG15. Every manifold of type K3[2] admitting an action of eGi for some i must necessarily have Picard rank 21. This fact allows us to use lattice-theoretic methods to classify all the finite groups G acting faithfully on a hyper-K ̈ahler fourfold of type K3[2] X such that G contains eGi as a proper subgroup and eGi acts symplectically on X. We also describe examples of fourfolds of K3[2]-type admitting an action of such group G.