Unsolvability of finite groups isospectral to the automorphism group of the second sporadic Janko group

For a finite group $G$, the spectrum is the set $\omega(G)$ of element orders of the group $G$. The spectrum of $G$ is closed under divisibility and is therefore uniquely determined by the set $\mu(G)$ consisting of elements of $\omega(G)$ that are maximal with respect to divisibility. We prove that a finite group isospectral to ${\rm Aut}(J_2)$ is unsolvable.