Let $N(G)$ be the set of conjugacy class sizes of finite group $G$. Let $G$ be a finite group for which $N(G)=N(A_n)$, where $A_n$ is the alternating group of degree $n$.
Let $p$ and $q$ be distinct prime numbers satisfying $n/2<p,q<n$. Moreover, assume that the Sylow $p$-subgroups and Sylow $q$-subgroups of $G$ are cyclic. Is it true to say if $x\in G$ is of order $p$, then $q\mid|x^G|$?