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)⊂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)∘H(s)∘G(t), (s,t)∈D(K), for some function p:D(K)→E, can be represented in terms of solutions of the corresponding generalized Cauchy equation.