  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, that is $L\leq G\leq \operatorname{Aut}L$; 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 ${}^3D_4(2)$.
