5.41 (1976)

Yes, it does (mod CFSG). Otherwise, if $G$ is a counterexample, the centralizer of the product $E$ of all minimal normal subgroups of $G$ is trivial and there is an element of $G/E$ whose order is $m$, where $m$ is the maximum of the orders of 2-elements of $G$. By Theorem 1 in (M. Aschbacher, P. B. Kleidman, M. W. Liebeck, Math. Z., 208, no. 3 (1991), 401–409), which is proved using CFSG, there is an element of order $2m$ in $G$, which contradicts the choice of $m$. (S. A. Syskin, Letter of August, 20, 1992.)