Commuting involutions in finite simple groups

We prove that if G is a finite simple group and \(x, y \in G\) $x,y\in G$ are involutions, then \(|x^G \,{\cap }\, C_G(y)| \rightarrow \infty \) $|x^G\cap C_G(y)|\to \infty$ as \(|G| \rightarrow \infty \) $|G|\to \infty$ . This extends results of Guralnick–Robinson and Skresanov. We also prove a related result about \(C_{G}(t)/O(C_G(t))\) $C_G(t)/O(C_G(t))$ that does not require the classification of finite simple groups.