Representation of group isomorphisms. The compact case

Let $G$ be a discrete group and let $\mathcal A$ and $\mathcal B$ be two subgroups of $G$-valued continuous functions defined on two $0$-dimensional compact spaces $X$ and $Y$. A group isomorphism $H$ defined between $\mathcal A$ and $\mathcal B$ is called \textit{separating} when for each pair of maps $f,g\in \mathcal A$ satisfying that $f^{-1}(e_G)\cup g^{-1}(e_G)=X$, it holds that $Hf^{-1}(e_G)\cup Hg^{-1}(e_G)=Y$. We prove that under some mild conditions every separating isomorphism $H:\mathcal A\longrightarrow \mathcal B$ can be represented by means of a continuous function $h: Y\longrightarrow X$ 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.