  Motivated by $\tau$-tilting theory developed by Adachi, Iyama and Reiten, for
a finite-dimensional algebra $\Lambda$ with action by a finite group $G$, we
introduce the notion of $G$-stable support $\tau$-tilting modules. Then we
establish bijections among $G$-stable support $\tau$-tilting modules over
$\Lambda$, $G$-stable two-term silting complexes in the homotopy category of
bounded complexes of finitely generated projective $\Lambda$-modules, and
$G$-stable functorially finite torsion classes in the category of finitely
generated left $\Lambda$-modules. In the case when $\Lambda$ is the
endomorphism of a $G$-stable cluster-tilting object $T$ over a Hom-finite
2-Calabi-Yau triangulated category $\mathcal{C}$ with a $G$-action, these are
also in bijection with $G$-stable cluster-tilting objects in $\mathcal{C}$.
Moreover, we investigate the relationship between stable support $\tau$-tilitng
modules over $\Lambda$ and the skew group algebra $\Lambda G$.
