Endo-permutation modules arising from the action of a p-group on a defect zero block

Let $p$ be an odd prime and let $k$ be an algebraically closed field of characteristic $p$.  Also, let $G$ be a $p'$-group.  Then by Maschke's theorem we know $kG\cong \prod_{i=1}^{t} \End_k(V_i)$ as $k$-algebras.  Suppose that a $p$-group $P\leq \Aut(G)$ stabilizes $\End_k(V_{i_0})$ for some $i_0$.  Such a $V_{i_0}$ will be an endo-permutation $kP$-module. Puig showed that the only modules that occur in this way are those whose image is torsion in the Dade group $D(P)$.

If we let $G$ be any finite group and let $b$ be a defect zero block of $kG$, then $kGb\cong\End_k(L)$ for some $L$.  If $kGb$ is $P$-stable for some $p$-group $P\leq \Aut(G)$ and $\Br_P(b)\neq 0$, then $L$ will again be an endo-permutation $kP$-module.  We show that if $p\geq 5$, then $L$ is torsion in $D(P)$.  This result depends on the classification of the finite simple groups.