Nearrings of Functions Determined by a Collection of Invariant Subgroups

Let (G, +) be a finite group, written additively with identity 0, but not necessarily abelian. Let Gamma = {H-i} be a nonempty collection of nonzero, proper subgroups of G. Then N = {f : G -> G | f (0) = 0 and f(H-i) subset of H-i for all i} is a nearring under pointwise addition and function composition. We determine when N is abelian and distributive, identify the center of N, and find necessary and sufficient conditions for the center to be a subnearring of N.