  H\"ohn and Mason classified the possible symplectic groups acting on an
Irreducible Holomorphic Symplectic (IHS) manifold of K3$^{[2]}$-type, finding
that $\mathbb Z_3^4 : \mathcal A_6$ is the symplectic group with the biggest
order. In this paper, we study the possible IHS manifolds of K3$^{[2]}$-type
with a symplectic action of $\mathbb Z_3^4 : \mathcal A_6$ and also admitting a
non-symplectic automorphism. We characterize such IHS manifolds. In particular
we prove that there exists a IHS manifold of K3$^{[2]}$-type with finite
automorphism group of order 174960, the biggest possible order for the
automorphism group of a IHS manifold of K3$^{[2]}$-type, and it is the Fano
variety of lines of the Fermat cubic fourfold.
