On the composite Pexider equation modulo a subgroup

Let X, Y, Z be arbitrary nonempty sets, E be a subgroup of the group of all bijections of Z (with composition of functions as the group operation), and K be a nonempty set with a binary operation defined on D(K) subset of K-2. Conditions are established under which functions F, G, H mapping K into Z(X), Y-X, Z(Y), resp., and satisfying the generalized composite Pexider equation F(st) = p(s, t) o H(s) o G(t), (s, t) E D(K), for some function p, : D(K) --> E, can be represented in terms of solutions of the corresponding generalized Cauchy equation.