Given a group$G$ and a representation$V$ of $G$, then any subgroup inclusion $H\hookrightarrow G$ makes also $H$act on $V$, this is the restricted representation.

Given an irreducible representation of $G$, then its decomposition as a direct sum of irreducible representations of $H$, after restricting the respesentation to $H$, is often called the “branching rule” of the restriction.

More generally, for any group homomorphism$H \to G$ representations of $G$ “pull back” to representations of $G$.