Classification of Finite Groups With Equal Left and Right Quotient Sets

In this paper, we classify all finite groups $G$ which have the following property: for all subsets $A\subseteq G$, we have $\mathrm{\mid }A{A}^{-1}\mathrm{\mid }=\mathrm{\mid }{A}^{-1}A\mathrm{\mid }$. This question is motivated by the problem in additive combinatorics of More Sums Than Difference sets and answers several questions posed in arXiv:2509.00611 [math.NT].