Herzog's conjecture on the number of involutions

Let $G$ and $H$ be simple groups, and let $I_2(G)$ and $I_2(H)$ denote their numbers of involutions.

Herzog's conjecture. If $I_2(G)=I_2(H)$, then $|G|=|H|$.

The conjecture was proposed by M. Herzog in 1979. The paper gives a counterexample, so the assertion is false.