This should be easy for the resident representation theory specialists:
Let $F$ be an algebraically closed field of characteristic $p>0$, $G$ a finite group, and $M$ an indecomposable $FG$-module that is a direct summand of $\mathrm{Ind}_{G/H} 1$. Is a vertex of $M$ given by a $p$-Sylow of $H$ or can it be smaller? I am sure that it can be smaller in general (an example would be appreciated). Are there conditions on $H$ that ensure that the vertex is not smaller (e.g. $p$-Sylow normal in $H$; I cannot seem to get Green correspondence to work for me)?
As a reminder, the vertex of $M$ is a minimal subgroup $U$ of $G$ with the property that $M$ is a direct summand of $\mathrm{Ind}_{G/U} 1$. It is always a $p$-group and is well-defined up to conjugation.