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.

notinduced from a proper subgroup? That happens only when $K \cap g^{-1}Hg=K$ for some $g$, i.e. when $K$ is subconjugate to $H$. $\endgroup$