Let $H$ be a normal subgroup of $G$. The action $\varphi_g$ of $G$ on $H$ by conjugation may not preserve a representation, because $h$ and $\varphi_g(h) = ghg^{-1}$ are not necessarily in the same conjugacy class of $H$ (even though they are in the same class in $G$).
Therefore, given an irreducible representation $\rho$ of $H$, we may have $\rho \circ \varphi_g$ different from $\rho$. Is it true that $\varphi_h$ for $h \in H$ "preserves" the representation $\rho$, in the sense that $\varphi_g \circ \rho$ is the same map as $\rho$?