In the mid 90's, Martino- Priddy proved that given two finite groups $G, H$, the following are equivalent:
- $\mathbb{F}_p\mathrm{Rep}(P,G)\cong \mathbb{F}_p\mathrm{Rep}(P,H)$ as $\mathbb{F}_p\mathrm{Out}(P)$-modules, for every finite $p$-group $P$.
- $BG^{\wedge}_{p}$ 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 $\{(BG_+)^{\wedge}_p\,(BH_+)^{\wedge}_p\}$