$PGL_2(q)$ cannot be determined by its $cs$

Abstract

For a finite group $G$, let $Z(G)$ denote the center of $G$ and $cs^{\ast}(G)$ be the set of non-trivial conjugacy class
        sizes of $G$. In this paper, we show that if $G$ is a finite group such that for some odd prime power $q \geq 4$,
        $cs^{\ast}(G)=cs^{\ast}(PGL_2(q))$, then either $G \cong PGL_2(q) \times Z(G)$ or $G$ contains a normal subgroup $N$
        and a non-trivial element
        $t \in G$ such that $N \cong PSL_2(q) \times Z(G)$, $t^2 \in N$ and $G = N . \langle t\rangle$.
        This shows that the almost simple groups cannot be
        determined by their set of conjugacy class sizes (up to an abelian direct
        factor).