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In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:

  1. $\mathbb{F}_p\mathrm{Inj}(P,G)\cong \mathbb{F}_p\mathrm{Inj}(P,H)$ as $\mathbb{F}_p\mathrm{Out}(P)$-modules, for every finite $p$-group $P$.
  2. $BG^{\wedge}_{p}$ and $BH^{\wedge}_p$ are stably homotopy equivalent.

Their proof made use of a matrix giving the multiplicity of each indecomposable stable summand, and needed other two equivalent conditions. By the time this result was achieved, Webb applied the theory of inflation functors to show some results on the stable splitting of $BG^{\wedge}_p$, a very elegant approach.

With the modern machinery (Mackey/biset functors, ghost algebras, fusion systems), is it possible to give a more direct proof?.

Edit: This is my attempt, let $\mathrm{Inj}(P,G)=\{[h]\in\mathrm{Rep}(P,G)\mid h \mathrm{\ is\ a\ monomorphism}\}$, we can can consider $M_P=\mathbb{F}_p\mathrm{Inj}(P,-)$ as a global Mackey functor (using Webb's terminology). Suppose that $M_P(G)\cong M_P(H)$ as $\mathbb{F}_p\mathrm{Out}(P)$-modules, for all finite $p$-group $P$.

According to A ghost ring for the left-free double Burnside ring and an application to fusion systems by Boltje-Danz, there is a monomorphism

$$\sigma_{G,H}:\mathbb{F}_pB^{\Delta_p}(G,H)\to\displaystyle\bigoplus_{P}{\mathrm{Hom}_{\mathbb{F}_p\mathrm{Out}(P)}(M_P(H),M_P(G))},$$ where $P$ runs over the classes of isomomorphisms of finite $p$-groups. Then, each simple $\Delta_p$-biset functor $S_{Q,V}$ is a composition factor of $M=\displaystyle\bigoplus_{P}{M_P}$ (seeing this latter as a $\Delta_p$-biset functor too), I expect somehow it implies that $\mathrm{dim}_{\mathbb{F}_p}S_{Q,V}(G)=\mathrm{dim}_{\mathbb{F}_p}S_{Q,V}(H)$ so that $\mathbb{F}_pB^{\Delta_p}(-,G)\cong\mathbb{F}_pB^{\Delta_p}(-,H)$, and finally $G$ is isomorphic to $H$ in the category $\mathbb{F}_pB^{\Delta_p}$.

Note that $\mathbb{F}_pB^{\Delta_p}(G,H)$ can be seen as a submodule of $\{(BG_+)^{\wedge}_p\,(BH_+)^{\wedge}_p\}\otimes\mathbb{F}_p$, I am not sure if any isomorphism in $\{(BG_+)^{\wedge}_p, (BH_+)^{\wedge}_p\}\otimes\mathbb{F}_p$ can be lifted to an isomorphism in $\{(BG_+)^{\wedge}_p\,(BH_+)^{\wedge}_p\}$

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    $\begingroup$ I understand that Martino and Priddy did not prove the implication (1)=>(2). A complete proof was given by Bob Oliver in a pair of papers: Equivalences of classifying spaces completed at odd primes. Math. Proc. Cambridge Philos. Soc. (2004) and Equivalences of classifying spaces completed at the prime two. Mem. Amer. Math. Soc. (2006). Oliver's proof used the classification of finite simple group and the obstruction theory of Broto-Levi-Oliver (cont.) $\endgroup$
    – Gregory Arone
    Commented Aug 21, 2021 at 8:55
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    $\begingroup$ Later Chermak gave another proof, (Acta Math. 211, 2013) which was again simplified by Oliver (Acta Math. 211, 2013). These proofs replaced the topological obstruction theory of BLO with a more algebraic approach. But AFAIK all proofs to date use the classification of finite simple groups. $\endgroup$
    – Gregory Arone
    Commented Aug 21, 2021 at 8:56
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    $\begingroup$ @GregoryArone you are referring to the unstable homotopy characterization, I guess. This is the stable one, there is no mistake in their proof so far (Ragnarsson found one, then it was corrected). $\endgroup$
    – Victor TC
    Commented Aug 23, 2021 at 16:30
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    $\begingroup$ Right, I missed that you are only asking about the proof of the result about stable homotopy. Sorry. $\endgroup$
    – Gregory Arone
    Commented Aug 23, 2021 at 16:42
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    $\begingroup$ Lifting that isomorphism is basic representation theory. $\endgroup$
    – Nicholas Kuhn
    Commented Oct 1, 2021 at 0:10

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