I have an elementary question about equivariant loop spaces that I feel it should be well known.
Given a finite group $G$ and a finite $G$-set $J$ let $S^J=\mathbb{R}[J]^+$ be the permutation representation sphere, and let $\Omega^J$ and $\Sigma^J$ be the corresponding loop and suspension functors on pointed $G$-spaces. If $\alpha\colon K\to J$ is a surjective $G$-map of finite $G$-sets, there is a suspension map $$\alpha^\ast\colon \Omega^J\Sigma^JX\to \Omega^K\Sigma^KX$$ It is defined by smashing with the sphere of the canonical orthogonal complement of the inclusion $\alpha^\ast\colon \mathbb{R}[J]\to \mathbb{R}[K]$. My question is:
$$\mbox{When does the restriction of $\alpha^\ast$ on $G$-fixed points } \alpha^\ast\colon (\Omega^J\Sigma^JX)^G\to (\Omega^K\Sigma^KX)^G \mbox{ have a retraction? }$$
I'd be happy to know what happens when $J$ and $K$ are transitive, as these retractions should give the Tom Dieck-splitting.
Here's what I know: if $\alpha\colon K\to K/H$ is the projection onto the quotient of $K$ by a normal subgroup $H\leq G$, a retraction exists. It is constructed as follows: $$(\Omega^K\Sigma^KX)^G=((\Omega^K\Sigma^KX)^H)^G\stackrel{res^H}{\longrightarrow}(\Omega^{K/H}\Sigma^{K/H}X^H)^G=(\Omega^{K/H}\Sigma^{K/H}X)^G$$ where the middle map restricts an equivariant loop to the $H$-fixed-points. We are using the canonical isomorphism $\mathbb{R}[K]^H\cong\mathbb{R}[K/H]$.
