Finite simple groups have many classes of $p$-elements

For an element $x$ of a finite group $T$, the $\mathrm{A}\mathrm{u}\mathrm{t}(T)$-class of $x$ is the set $\{x^{\sigma }\mid \sigma \in \mathrm{A}\mathrm{u}\mathrm{t}(T)\}$. We prove that the order $\mathrm{\mid }T\mathrm{\mid }$ of a finite nonabelian simple group $T$ is bounded above by a function of the parameter $m(T)$, where $m(T)$ is the maximum, over all primes $p$, of the number of $\mathrm{A}\mathrm{u}\mathrm{t}(T)$-classes of elements of $T$ of $p$-power order. This bound is a substantial generalisation of results of Pyber, and of Héthelyi and Külshammer, and it has implications for relative Brauer groups of finite extensions of global fields.