On $p$-parts of conjugacy class sizes and the index of the $p$-core

For a prime $p$ and an arbitrary finite group $G$, we show that if $p^2$ does not divide the size of each conjugacy class of \emph{$p$-regular} element (element of order not divisible by $p$) in $G$, then the largest power of $p$ dividing the index $\mathrm{\mid }G:{\mathbf{O}}_p(G)\mathrm{\mid }$ is at most $p^2$.