  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
$\mathrm{Ind}_G^{\tilde{G}}M$ to be a support $\tau$-tilting
$k\tilde{G}$-module for a $kG$-module $M$. Moreover, we give the block version
of the result.
