Irreducible Holomorphic Symplectic manifolds with an action of ${\mathbb{Z}}_3^4:{\mathcal{A}}_6$

Höhn 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.