The author defines the notion of (h,A,B)-linear map between two groups G and G′ (where A⊆G,B⊆G′). This generalizes the notion of λ-linear map introduced in the author's earlier papers. Some properties of the (h,A,B)-linear maps are given, together with a theorem of existence and uniqueness of such a map (under certain conditions on h,A,B). An application in cryptography with a concrete example is presented. The group Λ(G_A) of the bijective (h,A)-applications, where h is surjective, is studied and described as a certain semi-direct product. The theory is applied to list all Λ(G_A)'s, for G the cyclic group with 6 elements and A⊆G. Finally, a generalization of the notion of (h,A,B)-linear map to the case of MA-hypergroupoids (also introduced by the author) is given.
