Finite simple groups have many classes of prime order elements

Let $T$ be a finite non-abelian simple group. Giudici, Morgan and Praeger have shown that the order of $T$ is bounded above by a function depending on the maximum number of $\mathrm{A}\mathrm{u}\mathrm{t}(T)$-classes of elements of $T$ of prime-power order. In this note, we strengthen this result by showing, in particular, that prime-power can be replaced by prime.