This paper is concerned with the structure of M = Maps(G), the near-ring of all mappings from a group G to itself which commute with a group S* of automorphisms of G. Here S is S* together with the zero endomorphism. Necessary and sufficient conditions on the pair (G, S) are obtained for M to be (i) regular, (ii) unit regular, (iii) an equivalence near-ring. These conditions take a very simple form. In the case (iii), the two-sided M-subgroups of M are determined. The next result shows that under suitable conditions, M is a simple near-ring. A definition of transitivity is given for subnear-rings of M, and some properties of transitive near-rings are proved. Finally two examples are given to show that all the classes of near-rings considered are distinct.