Let $K$ be a compact Lie group. Let $C_K$ denote the category whose objects are the compact lie groups containing $K$ and whose morphisms are inclusion of the groups. Let $Y$ be a $K-$space such that $Y^L$ is contractible for each closed subgroup $L$ of $K$ except $K$ itself and $Y^K$ is empty.

Let $H\leqslant G$ be two objects in $C_K$. By equivariant obstruction theory, each $K-$map $$f: H\longrightarrow Y$$ can be extended to a $K-$map $$\widetilde{f}: G\longrightarrow Y$$ which is unique up to homotopy rel $H$.

I wonder whether there is any specific construction of the extension $\widetilde{f}$ such that we have a covariant functor from $C_K$ to the category of topological spaces, sending $G$ to $Map_K(G, Y)$, and a morphism $H\hookrightarrow G$ to $f\mapsto \widetilde{f}$.

Obviously, we have a contravariant functor from $C_K$ to the category of topological spaces, sending $G$ to $Map_K(G, Y)$, and a morphism $H\hookrightarrow G$ to $g\mapsto g|_H$ where $g\in Map_K(G, Y)$ and $g|_H$ is its restriction to $H$.

But I'm thinking about the other direction and don't have good idea how to find such a "standard" extension $\widetilde{f}$.