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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 mainresult in this paper is to show that support $\tau$-tilting$k\tilde{G}$-modules satisfying some properties are support $\tau$-tiltingmodules as $kG$-modules too. As the second main result, we give equivalentconditions for support $\tau$-tilting $k\tilde{G}$-modules to satisfy the aboveproperties, 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 orderedsets. 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 ofindecomposable $\tau$-rigid $k\tilde{G}$-modules. Finally, we give the blockversions of the above results.

Normal subgroups and support $\tau$-tilting modules