Given a locally compact (topological) group $G$ we denote by $C_b^{lu}(G)\subseteq C_b(G)$ the bounded continuous functions on $G$ such that $f\in C_b^{lu}(G)$ whenever the map $G\to C_b(G)$ given by $g\mapsto g.f$ is continuous. Here, the action is given by left-translation, i.e. $g.f(x)=f(g^{-1}x)$, and $C_b(G)$ is equipped with the sup-norm.
It is easy to see that $C_b^{lu}(G)$ is a unital and commutative $C^*$-algebra and so is isomorphic to $C_0(X)$ for some compact (Hausdorff) space $X$. One can prove that we have a dense and open embedding $G\subseteq X$ settling that $X$ is indeed a compactification of $G$.
Now, let $Y$ denote a compact (Hausdorff) space equipped with a (strongly) continuous group action by $G$. Fixing $y_0\in Y$, I have seen it claimed that the continuous and $G$-equivariant map $G\to Y$ given by $g\mapsto g.y_0$ extends to a continuous and $G$-equivariant map $X\to Y$. Why is this the case?
More generally, can one describe precisely which continuous maps into compact spaces that admit such an extension?
(Of course, if $G$ were discrete, then $X$ would be the Stone-Cech compactification and so the result falls out immediately).
Thank you!