Lipschitz metric isometries between outer spaces of virtually free groups

Dowdall and Taylor observed that, given a finite-index subgroup of a free group, taking covers induces an embedding from the outer space of the free group to the outer space of the subgroup, this embedding is an isometry with respect to the (asymmetric) Lipschitz metric and that the embedding sends folding paths to folding paths. The purpose of this note is to extend this result to virtually free groups. We further extend a result of Francaviglia and Martino, proving the existence of "candidates" for the Lipschitz distance between points in the outer space of the virtually free group. Additionally, we identify a deformation retraction of the spine of the outer space for the virtually free group with the space considered by Krstie and Vogtmann.