Abelian sharp permutation groups

A permutation group G of finite degree n is a sharp irredundant group of type {k}, k a positive integer, if each non-identity element of G fixes exactly k points, |G|=n−k and G has no global fixed point and no regular orbit. In this note we give a method to construct all faithful representations of finite abelian groups as sharp irredundant permutation groups of type {k} for some positive integer k.