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

1. $\mathbb{F}_p\mathrm{Rep}(P,G)\cong \mathbb{F}_p\mathrm{Rep}(P,H)$ as $\mathbb{F}_p\mathrm{Out}(P)$-modules, for every finite $p$-group $P$.
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?.