On the generation of finite groups by classes of involutions

Let $D$ be an invariant subset of involutions of the finite group $G$. $D$ satisfies the condition of coherence, if for any two distinct commuting involutions of $D$ their product also belongs to $D$. $D$ satisfies the condition of separability if the product of any two involutions of is a 2-element or a $2'$-element.
In this paper it is proved that if the finite group $G$ is generated by an invariant subset of involutions $D$ satisfying the coherence and separability conditions, and if $D\cap O_2(G)=\varnothing$, then either $G$ has a Sylow 2-subgroup of order 2, or $Z(G)$ has odd order, $G=G'$, and the factor group $G/Z(G)$ is isomorphic to one of the following simple groups: $L_2(p)$, $p$ a Fermat or a Mersenne prime number, $L_2(q)$, $Sz(q)$, $U_3(q)$, $L_3(q)$, $G_2(q)$ ($G_2(q)'$ respectively), $^3D_4(q)$, $q$ even, $A_6$ or $J_2$.
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