Bouc's conjecture on $B$-groups

Bouc proposed the following conjecture: a finite group $G$ is nilpotent if and only if its largest quotient $B$-group $\beta (G)$ is nilpotent. And he has prove that this conjecture holds when $G$ is solvable. In this paper, we consider the case when $G$ is not solvable. Let $S$ be a nonabelian simple group except the Chevalley groups $A_n(q)$, $D_n(q)$, $E_6(q)$, and ${}^2{A}_n(q)$, if there exists only one factor of $G$ which is isomorphic to $S$, then $\beta (G)$ is not solvable, of course, is not nilpotent. That means we prove the conjecture in these cases.