The binary actions of simple groups with a single conjugacy class of involutions

We continue our investigation of binary actions of simple groups. In this paper, we demonstrate a connection between the graph $\mathrm{\Gamma }(\mathcal{C})$ based on the conjugacy class $\mathcal{C}$ of the group $G$, which was introduced in our previous work, and the notion of a strongly embedded subgroup of $G$. We exploit this connection to prove a result concerning the binary actions of finite simple groups that contain a single conjugacy class of involutions.