  We give a sufficient condition for the $kG$-Scott module with vertex $P$ to
remain indecomposable under taking the Brauer construction for any subgroup $Q$
of $P$ as $k[Q\,C_G(Q)]$-module, where $k$ is a field of characteristic $2$,
and $P$ is a wreathed $2$-subgroup of a finite group $G$. This generalizes
results for the cases where $P$ is abelian and some others. The motivation of
this paper is that the Brauer indecomposability of a $p$-permutation bimodule
($p$ is a prime) is one of the key steps in order to obtain a splendid stable
equivalence of Morita type by making use of the gluing method that then can
possibly lift to a splendid derived equivalence.
