Let $G$ be a finite group and $N$ a normal subgroup. One of the modern ways to construct the $\infty$-category of $G$-spectra is as product-preserving spectral presheaves $\text{Sp}^G = \text{Fun}^{\times}(A^{\text{eff}}(\mathcal{O}_G^{\sqcup}), \text{Sp})$ where $A^{\text{eff}}$ is the effective Burnside category construction of Barwick and $\mathcal{O}_G^{\sqcup}$ is the finite coproduct completion of the orbit category $\mathcal{O}_G$.

As far as I understand (and maybe I'm already mistaken here) the way to pass from $G$-spectra to $G/N$-spectra via the genuine fixed points is to consider the inclusion $j_N : \mathcal{O}_{G/N} \hookrightarrow \mathcal{O}_{G}$ which induces $(-)^N := j_N^* : \text{Fun}^{\times}(A^{\text{eff}}(\mathcal{O}_G^{\sqcup}), \text{Sp}) \rightarrow \text{Fun}^{\times}(A^{\text{eff}}(\mathcal{O}_{G/N}^{\sqcup}), \text{Sp})$, that is, we're just keeping the genuine fixed point data below the orbit $G/N$ in the diagram.

On the other hand, for $X \in \text{Sp}^G$, we can also consider the geometric fixed points $\Phi^NX \in \text{Sp}^{G/N}$ which can be taken to be $\big(\widetilde{E\mathscr{F}[N]}\otimes X\big)^N$ where $\mathscr{F}[N] = \{K \leq G \: | \: N \not \leq K\}$. But then a little manipulation with the Wirthmuller isomorphism and the corepresentability of the fixed point data gives us \begin{equation*} \big(\widetilde{E\mathscr{F}[N]}\otimes X\big)^K = \begin{cases} 0 & \text{ if } K \in \mathscr{F}[N]\\ X^K & \text{ if } K \not \in \mathscr{F}[N] \end{cases} \end{equation*} This can be found for example as Lemma A.18 in Denis Nardin's thesis.

Question: This seems to say that $X^N \simeq \big(\widetilde{E\mathscr{F}[N]}\otimes X\big)^N \in \text{Sp}^{G/N}$ (which of course shouldn't be true), since both sides are just constructions that keep the fixed points data below the orbit $G/N$ by the arguments above. What went wrong?

I feel like the problem lies with my understanding of the smashing localisation. The construction of the geometric fixed points as some left Kan extension shows us how the data from the family $\mathscr{F}[N]$ is used to compute the geometric fixed points (as we're taking colimits involving orbits coming from this family; think Brauer quotients in ordinary Mackey functors), whereas $\widetilde{E\mathscr{F}[N]}\otimes-$ as understood above seems to just ignore these information.


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