Almost recognizability by spectrum of simple exceptional groups of Lie type

The spectrum of a finite group is the set of its elements orders. Groups are said to be isospectral if their spectra coincide. For every finite simple exceptional group $L=E_7(q)$, we prove that each finite group isospectral to $L$ is isomorphic to a group $G$ squeezed between $L$ and its automorphism group, i.e., $L\le G\le\mathrm{Aut}\,L$; in particular, up to isomorphism, there are only finitely many such groups. This assertion, together with a series of previously obtained results, implies that the same is true for every finite simple exceptional group except the group $^3D_4(2)$.