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

Irreducible Holomorphic Symplectic manifolds with an action of $\mathbb Z^4_3 : \mathcal A_6$