  G. H\"ohn and G. Mason classified all finite groups acting faithfully and
symplectically on a hyper-K{\"a}hler fourfolds of type K3$^{[2]}$. There are 15
maximal among them, call them $\widetilde{G}_1,\ldots, \widetilde{G}_{15}$.
Every manifold of type K3$^{[2]}$ admitting an action of $\widetilde{G}_i$ 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{\"a}hler fourfold of type K3$^{[2]}$ $X$
such that $G$ contains $\widetilde{G}_i$ as a proper subgroup and
$\widetilde{G}_i$ acts symplectically on $X$. We also describe examples of
fourfolds of K3$^{[2]}$-type admitting an action of such groups.
