Counterexamples on the fusion of involutions in finite groups

Abstract:

Some examples are given of simple groups $G$ containing a conjugacy class of involutions $\mathcal {C}$ such that if $S$ is a Sylow $2$-subgroup of $G$, then there exist subgroups $A \subseteq B \subseteq S$ with $|A:B| = 2$ and $\mathcal {C} \cap S \subseteq B - A$.