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We present a sufficient condition for the $kG$
-Scott module with vertex $P$
to remain indecomposable under 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 semidihedral $2$
-subgroup of a finite group $G$
. This generalizes results for the cases where $P$
is abelian or dihedral. The Brauer indecomposability is defined by R. Kessar, N. Kunugi and N. Mitsuhashi. The motivation of this paper is the fact that the Brauer indecomposability of a $p$
-permutation bimodule (where $p$
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 due to Broué, Rickard, Linckelmann and Rouquier, that then can possibly be lifted to a splendid derived (splendid Morita) equivalence.
