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 $M^{c}L$, $M_{12}$, $M_{22}$, $He$, $Suz$, $O^{\mathrm{\prime }}N$, 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 $\mathrm{A}\mathrm{u}\mathrm{t}(J_2)$, then either $G$ is isomorphic to $\mathrm{A}\mathrm{u}\mathrm{t}(J_2)$, or $G$ is an extension of a $2$-group by ${\mathbb{A}}_8$.