Some properties that are preserved by transferring boundary functors

If a Hausdorff locally compact paracompact space has a coarse structure, then there is a family of well behaved compactifications associated to it. If there are two of these spaces, $X$ and $Y$, with a good coarse equivalence, then there is a correspondence between these families of compactifications of $X$ and $Y$. On the other hand, if a group $G$ has a properly discontinuous cocompact action on a Hausdorff locally space $X$, then there is also a correspondence between nice compactifications of $G$ and nice compactifications of $X$. In this paper we show that when there are both concepts involved (coarse structure and group action), then both correspondences of families of compactifications agree. We also prove that these correspondences must preserve some geometric properties of the compactifications.