Quasirecognizability by the Set of Element Orders for Groups $^3D_4(q)$ and $F_4(q)$, for $q$ Odd

It is proved that if $L$ is one of the simple groups $^3D_4(q)$ or $F_4(q)$, where $q$ is odd, and $G$ is a finite group with the set of element orders as in $L$, then the derived subgroup of $G/F(G)$ is isomorphic to $L$ and the factor group $G/G'$ is a cyclic $\{2,3\}$-group.