Homotopy groups of certain geometric fixed point spectrum

Let $$G$$ be a finite group and $$E$$ be a genuine $$H$$-spectrum for $$H\leq G.$$ Then for any subgroup $$K$$ of $$G$$, consider the $$K$$-spectrum $$X=Res^G_K Ind^G_H(E).$$

Is there any reference for computing the homotopy groups of the geometric fixed point $$\Phi^K(X): = (\widetilde{E\mathcal{P}}\wedge X)^K?$$ Here $$\widetilde{E\mathcal{P}}$$ is a $$K$$-space given the following fixed point data:

$$\widetilde{E\mathcal{P}}^L= \begin{cases} S^0, & \text{ if }L=K\\ \ast, & \text { if } L \text{ is any proper subgroup of } K.\end{cases}$$ Maybe the double coset formula could be useful.

Thank you so much in advance. Any help will be appreciated.

• back of the envelope guess: if $K$ is not subconjugate to $H$ then you should get 0, if $K$ is subconjugate to $H$ then you should end up with $|G/H|$-copies of $\Phi^K(E)$, I think. Does that sound right? Sep 14, 2020 at 12:11
• @DylanWilson: Yes, it sounds interesting. Can you please write down the key arguments for it? Sep 14, 2020 at 12:24
• As you said, I think it's the double coset formula. $\Phi^K$ annihilates anything induced from a proper subgroup of $K$, so you just have to ask- when is one of the terms in the double coset formula not induced from a proper subgroup? That happens only when $K \cap g^{-1}Hg=K$ for some $g$, i.e. when $K$ is subconjugate to $H$. Sep 14, 2020 at 12:43