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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">C</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> Γ(C) based on the conjugacy class 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.