Powers in the wreath product of $G$ with $S_n$

In this paper we compute powers in the wreath product $G\wr S_n$, for any finite group $G$. For $r\ge 2$, a prime, consider ${\omega }_r:G\wr S_n\to G\wr S_n$ defined by $g\mapsto g^r$. Let $P_r(G\wr S_n)=\frac{\mathrm{\mid }{\omega }_r(G\wr S_n)\mathrm{\mid }}{\mathrm{\mid }G{\mathrm{\mid }}^{n}n!}$, be the probability that a randomly chosen element in $G\wr S_n$ is a $r^{th}$ power. We prove, $P_r(G\wr S_{n+1})=P_r(G\wr S_n)$ for all $n\equiv ̸-1(\text{mod }r)$ if, order of $G$ is coprime to $r$. We also give a formula for the number of conjugacy classes that are $r^{th}$ powers in $G\wr S_n$.