Inducing $π$-partial characters with a given vertex

Let $G$ be a solvable group. Let $p$ be a prime and let $Q$ be a $p$-subgroup of a subgroup $V$. Suppose $ϕ\in \ibr G$. If either $|G|$ is odd or $p = 2$, we prove that the number of Brauer characters of $H$ inducing $ϕ$ with vertex $Q$ is at most $|\norm GQ: \norm VQ|$.

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