A set theoretic version of equations on groups

Let $G$ be a finite group. The aim of this paper is to study the number of solutions $S\subseteq G$ of the equation ${\mathrm{\mho }}^{\{n\}}(S)=L$, where $L$ is a non-empty subset of $G$, $n$ is a positive integer and ${\mathrm{\mho }}^{\{n\}}(S)=\{s^n\mathrm{\mid }s\in S\}$. Besides our findings obtained in this general frame, we also outline some results which hold for some particular cases such as: $\mathit{\text{i)}}$ $L$ is a normal subset of $G$; $\mathit{\text{ii)}}$ $G$ is abelian; $\mathit{\text{iii)}}$ $G$ is an extraspecial $p$-group.