Quasirecognition by the set of element orders of the groups $E_6(q)$ and $^2E_6(q)$

We prove that if $L$ is one of the simple groups $^2E_6(q)$ and $E_6(q)$ and $G$ is some finite group with the same spectrum as $L$, then the commutant of $G/F(G)$ is isomorphic to $L$ and the quotient $G/G'$ is a cyclic $\{2,3\}$-group.