  For a discrete metric space (or more generally a large scale space) $X$ and
an action of a group $G$ on $X$ by coarse equivalences, we define a type of
coarse quotient space $X_G$, which agrees up to coarse equivalence with the
orbit space $X/G$ when $G$ is finite. We then restrict our attention to what we
call coarsely discontinuous actions and show that for such actions the group
$G$ can be recovered as an appropriately defined automorphism group
$\mathsf{Aut}(X/X_G)$ when $X$ satisfies a large scale connectedness condition.
We show that for a coarsely discontinuous action of a countable group $G$ on a
discrete bounded geometry metric space $X$ there is a relation between the
maximal Roe algebras of $X$ and $X_G$, namely that there is a
$\ast$-isomorphism $C^\ast_{\max}(X_G)/\mathcal{K} \cong
C^\ast_{\max}(X)/\mathcal{K} \rtimes G$, where $\mathcal{K}$ is the ideal of
compact operators. If $X$ has Property A and $G$ is amenable, then we show that
$X_G$ has Property A, and thus the maximal Roe algebra and full crossed product
can be replaced by the usual Roe algebra and reduced crossed product
respectively in the above equation.
