Skew-product for group-valued edge labellings of Bratteli diagrams

Group valued edge labellings $\lambda$ of a Bratteli diagram $B$ give rise to a skew-product Bratteli diagram $B(\lambda)$ on which the group acts. The quotient by the group action of the associated dynamics can be a nontrivial extension of the dynamics of $B$. We exhibit a Bratteli diagram for this quotient and construct a morphism to $B$ with unique path lifting property. This is shown to be an isomorphism for the dynamics if a property `loops lifting to loops' is satisfied by $B(\lambda)\to B$.