Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space $$Map_{C_G(g)}(G, Y)^H,$$ the fixed point space of $Map_{C_G(g)}(G, Y)$ by $H$. 

I studied the image each double coset $C_G(g)\alpha H$. All the equivariant properties of the maps in $Map_{C_G(g)}(G, Y)$ (but may not be continous) if and only if the image of each $\alpha$ is in $Y^{\alpha H\alpha^{-1}\cap C_G(g)}$. I don't have a good idea how to put all these pieces together regarding the topology on $G$ to make it continuous map.

One question is: when is this space  $Map_{C_G(g)}(G, Y)^H$ nonempty even if each $Y^{\alpha H\alpha^{-1}\cap C_G(g)}$ is nonempty? What additional properties should it satisfy?

Another questions: Let $X$ be a $G-$subspace of $G$, which is the union of several double cosets $C_G(g)\alpha H$. Given a map in $Map_{C_G(g)}(X, Y)^H$, when can it be extended to a map in $Map_{C_G(g)}(G, Y)^H$? 
I have no good idea so far.

Thanks.