Representation of group isomorphisms. The compact case

Let.. be a discrete group and let A and B be two subgroups of G-valued continuous functions defined on two 0-dimensional compact spaces X and Y. A group isomorphism.. defined between A and B is called separating when, for each pair of maps f, g is an element of A satisfying that f(-1)(e(G)) boolean OR g(-1)(e(G)) = X, it holds that Hf-1(e(G)) boolean OR Hg-1(e(G)) = Y. We prove that under some mild conditions every biseparating isomorphism H : A -> B can be represented by means of a continuous function h : Y -> X as a weighted composition operator. As a consequence we establish the equivalence of two subgroups of continuous functions if there is a biseparating isomorphism defined between them.