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⊆ G S ⊆ G of the equation ^\(S)=L ℧ { n } ( S ) = L , where L is a non-empty subset of G , n is a positive integer and ^\(S)=\ sⁿ \ | \ s∈ S\ ℧ { n } ( S ) = { s n | s ∈ S } . Besides our findings obtained in this general frame, we also outline some results which hold for some particular cases such as: (i) L is a normal subset of G ; (ii) G is abelian; (iii) G is an extraspecial p -group.