The doubling-followed-by-rotation trick to prove the extremality of Gaussian distributions has been a valuable tool in information theory. In particular, the above trick has been used to establish the Gaussian extremality of a family of inequalities that unifies the Entropy Power Inequality and the Brascamp-Lieb inequalities. Here, we develop a technique (similar to the one in the continuous case) to prove the extremality of Haar distributions for a similar family of inequalities in finite Abelian groups.