Spectra of automorphic extensions of finite simple exceptional groups of Lie type

The spectrum $\omega(G)$ of a finite group $G$ is the set of orders of elements of $G$. Let $S$ be a simple exceptional group of type $E_6$ or $E_7$. We describe all finite groups $G$ such that $S\le G\le\operatorname{Aut}S$ and $\omega(G)=\omega(S)$. Along with the previously obtained results, this provides a description of all finite groups $G$ such that $\omega(G)=\omega(S)$ and completes the study of the recognition-by-spectrum problem for all simple exceptional groups of Lie type.