A sectional characterization of the Dade group

Let $k$ be a field of characteristic $p$, let $P$ be a finite $p$- group, where $p$ is an odd prime, and let $D(P)$ be the Dade group of endo-permutation $kP$-modules. It is known that $D(P)$ is detected via deflation–restriction by the family of all sections of $P$ which are elementary abelian of rank $\le 2$. In this paper, we improve this result by characterizing $D(P)$ as the limit (with respect to deflation–restriction maps and conjugation maps) of all groups $D(T\mathrm{/}S)$ where $T\mathrm{/}S$ runs through all sections of $P$ which are either elementary abelian of rank $\le 3$ or extraspecial of order $p^3$ and exponent $p$.