  We consider a finite dimensional strongly $G$-graded algebra $A$ with {
self-injective} $1$-component $B$, and in our main result we prove that the
induction from $B$ to $A$ of a basic support $\tau$-tilting pair of $B$-modules
is a support $\tau$-tilting pair $(M,P)$ of $A$-modules if and only if $M$ is
$G$-invariant. A similar statement holds for the restriction from $A$ to $B$,
so our results may be viewed as Clifford and Maschke type theorems for $2$-term
silting complexes. We also give applications to semibricks and the associated
wide subcategories.
