Normal subgroups and support $\tau$-tilting modules

Let $\tilde{G}$ be a finite group, $G$ a normal subgroup of $\tilde{G}$ and $k$ an algebraically closed field of characteristic $p>0$. The first main result in this paper is to show that support $\tau$-tilting $k\tilde{G}$-modules satisfying some properties are support $\tau$-tilting modules as $kG$-modules too. As the second main result, we give equivalent conditions for support $\tau$-tilting $k\tilde{G}$-modules to satisfy the above properties, and show that the set of the support $\tau$-tilting $k\tilde{G}$-modules with the properties is isomorphic to the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules as partially ordered sets. As an application, we show that the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules is isomorphic to the set of support $\tau$-tilting $k\tilde{G}$-modules in the case that the index $G$ in $\tilde{G}$ is a $p$-power. As a further application, we give a feature of vertices of indecomposable $\tau$-rigid $k\tilde{G}$-modules. Finally, we give the block versions of the above results.