  Coarse geometry, and in particular coarse homotopy theory, has proven to be a
powerful tool for approaching problems in geometric group theory and higher
index theory. In this paper, we continue to develop theory in this area by
proving a Coarse Lifting Lemma with respect to a certain class of bornologous
surjective maps. This class is wide enough to include quotients by coarsely
discontinuous group actions, which allows us to obtain results concerning the
coarse fundamental group of quotients which are analogous to classical
topological results for the fundamental group. As an application, we compute
the fundamental group of metric cones over negatively curved compact Riemannian
manifolds.
