Multiplicities of conjugacy class sizes of finite groups

It has been proved recently by Moreto and Craven that the order of a finite group is bounded in terms of the largest multiplicity of its irreducible character degrees. A conjugacy class version of this result was proved for solvable groups by Zaikin-Zapirain. In this note, we prove that if $G$ is a finite simple group then the order of $G$, denoted by $\mathrm{\mid }G\mathrm{\mid }$, is bounded in terms of the largest multiplicity of its conjugacy class sizes and that if the largest multiplicity of conjugacy class sizes of any quotient of a finite group $G$ is $m$, then $\mathrm{\mid }G\mathrm{\mid }$ is bounded in terms of $m$.