Induction preserves weak equivalences

Question

Let $G$ be a finite group and $H \leq G$ be a subgroup. There is an induction functor $G \ltimes_H - : \mathbf{Sp}^H \to \mathbf{Sp}^G$ from the category of $H$-spectra to the category of $G$-spectra (I preferably work with the model of orthogonal spectra) defined by the levelwise induction.

I want to understand the following

Claim. The functor $G \ltimes_H - : \mathbf{Sp}^H \to \mathbf{Sp}^G$ preserves $\underline{\pi}_{\bullet}$-isomorphisms.

So it sends $\underline{\pi}_{\bullet}$-isomorphisms to maps inducing isomorphisms on $\pi_{\bullet}^G$ by the Wirthmüller isomorphism. (Similarly for the coinduction). Now for instance this paper (Theorem 6.16) states that the claim follows from the double coset formula. Stefan Schwede employs a different argument in his Global Homotopy Theory but in a lecture he also said the same thing: It would at that point follow from a double coset argument.

I'd like to try certain naturalities and splittings involved in the double coset formula but right now I'm not sure how it should help.

So how exactly does the double coset formula help here?

Answer 1

I got an answer from Phil on discord.

Let $f:X \to Y$ be a $\underline{\pi}_{\bullet}$-isomorphism of $H$-spectra and let $K \leq G$ be a subgroup. We want to show that $f_*: \pi_{\bullet}^K(G \ltimes_H X) \to \pi_{\bullet}^K(G \ltimes_H Y)$ is an isomorphism. There is a splitting $$ \pi_{\bullet}^K(G \ltimes_H X) \cong \bigoplus_{KgH} \pi_{\bullet}^K(K \ltimes_{K \cap {}^g H} \operatorname{res}_{K \cap {}^g H}^{{}^g H} c_g^*X)$$ induced by a levelwise homeomorphism $\sum_{KgH} \kappa_g : \bigvee_{KgH} K \ltimes_{K \cap {}^g H} \operatorname{res}_{K \cap {}^g H}^{{}^g H} c_g^*X \to G \ltimes_H X$.

But then $f_*$ factors as a sum of maps $\pi_{\bullet}^K(K \ltimes_{K \cap {}^g H} \operatorname{res}_{K \cap {}^g H}^{{}^g H} c_g^*X) \to \pi_{\bullet}^K(K \ltimes_{K \cap {}^g H} \operatorname{res}_{K \cap {}^g H}^{{}^g H} c_g^*Y)$ and these are induced from $\pi_{\bullet}^{K^g \cap H}(X) \to \pi_{\bullet}^{K^g \cap H}(Y)$ by $c_g^*$ and the external transfer. This is an isomorphism since $f$ is a $\underline{\pi}_{\bullet}$-isomorphism. So the direct sum is also an isomorphism.