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