A characterization for induced modules to be support $\tau$-tilting modules

Let $\tilde{G}$ be a finite group and $G$ a normal subgroup of $\tilde{G}$. In this paper, we give a necessary and sufficient condition for ${\mathrm{I}\mathrm{n}\mathrm{d}}_G^{\tilde{G}}M$ to be a support $\tau$-tilting $k\tilde{G}$-module for a $kG$-module $M$. Moreover, we give the block version of the result.