Lifting Brauer indecomposability of a Scott module

It is proven that if a finite group $G$ has a normal subgroup $H$ with $p'$-index (where $p$ is a prime) and $G/H$ is solvable, then for a $p$-subgroup $P$ of $H$, if the Scott $kH$-module with vertex $P$ is Brauer indecomposable, then so is the Scott $kG$-module with vertex $P$, where $k$ is a field of characteristic $p>0$. This has several applications.