I was reading the paper "The Balmer spectrum of rational equivariant cohomology theories" of J.P.C. Greenlees and I found the following interesting fact, expressed in Lemma 4.2 and Remark 4.3.

Let $G$ be a compact Lie group and suppose we have a nesting of closed subgroups $G \geq H \geq K$. First we can show that the conjugacy classes of $K$ in $H$ are finitely many, let us denote by $K_i=K^{\gamma_i}$ for $i=0,\dots, n$ fixed representatives of such conjugacy classes (we can choose $\gamma_0=e$). Corresponding to this decomposition we have \begin{equation} \Phi^{K}(G_+\wedge_H Y)\cong \bigvee_{i=0}^n \gamma_i \Phi^{K_i}Y \end{equation} for a generic $H$-spectrum $Y$.

Where $\Phi^K X$ indicates the usual geometric fixed point spectrum of a $G$-spectrum $X$ with the residual $N_G(K)/K$-action, and $G_+ \wedge_H -$ is the induction functor left adjoint to the restriction from $G$-spectra to the $H$-spectra.

This would provide a general formula to compute the geometric fixed points of an induction spectrum, which is really useful. From what I understand it should work for generic compact Lie groups, without assumption of finiteness. Unfortunately, neither a reference of a proof is provided.

My first question is if this formula is true and you can provide references for it, or if it is false. If you can provide other formulas for computing the geometric fixed points of the induction of a spectrum I would be very interested, so feel free to share any similar result.

To make a reality check I tried to apply this formula to a non-trivial case. If we consider $H$ to be a finite cyclic group we know that the rational Burnside ring $\pi_0^H(S^0)$ is isomorphic to $C(\text{Sub}(H), \mathbb{Q})$, the ring of functions from the set of subgroups of $H$ to the rational numbers. Since the isomorphism is provided by the geometric fixed points, we deduce the existence of an idempotent $e_H$ such that $\Phi^K(e_HS^0)\simeq^e0$ if $K <H$ while $\Phi^H(e_HS^0)\simeq^e S^0$. Here the symbol $\simeq^e$ stands to remark that the equivalence presented is non-equivariant.

We now consider $H$ as a subgroup of the circle $S^1$ and set $\sigma_H=S^1_+ \wedge_H e_HS^0$, this is the basic $H$-cell of the $S^1$-equivariant rational spectra. They are the basic building block of the $S^1$-equivariant stable homotopy category and have been studied extensively in the literature, my main reference is the AMS monography "Rational $S^1$-equivariant stable homotopy theory" written by the same Greenlees.

If we were to apply the formula explained at the start we would get that non-equivariantly $\Phi^H\sigma_H$ is just $S^0$, but in the book I mentioned Lemma 2.1.3 shows that actually $\Phi^H \sigma_H \simeq^e S^0 \vee S^1$.

I really do not understand the reason of the discrepancy. Is the formula wrong or am I misinterpreting something?