Recognizing by Spectrum for the Automorphism Groups of Sporadic Simple Groups

The spectrum of a finite group is the set of its element orders, and two groups are said to be isospectral if they have the same spectra. A finite group $G$ is said to be recognizable by spectrum, if every finite group isospectral with $G$ is isomorphic to $G$. We prove that if $S$ is any of the sporadic simple groups $M^cL$, $M_{12}$, $M_{22}$, $He$, $Suz$, $O'N$, then ${\rm Aut}(S)$ is recognizable by spectrum. This finishes the proof of the recognizability by spectrum of the automorphism groups of all sporadic simple groups, except $J_2$. Furthermore, we show that if $G$ is isospectral with ${\rm Aut}(J_2)$, then either $G$ is isomorphic to ${\rm Aut}(J_2)$, or $G$ is an extension of a $2$-group by $\mathbb{A}_8$.