  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.
