Retraction theorems in group-compactifications.

Let $\Gamma$ be a discrete countable infinite Group and let $X$ be compact minimal $\Gamma$-space. A $\Gamma$-compactification by $X$ is a compact topology on $\Gamma\cup X$ on which $\Gamma$ acts continuously by left multiplication and the original action on $X$ respectively, and such that $\Gamma$ is dense in $\Gamma\cup X$.

Is there more than one way to “glue” $X$ to $\Gamma$ in such a way? Are there canonical families of $\Gamma$ compactifications? and what all of this has to do with the old and famous Brouwer’s non-retract theorem from classical topology?

Based on a joint work with Yair Hartman, Aranka Hrušková & Mehrdad Kalantar