Suppose $G$ is a compact topological group and $H$ is a closed subgroup whose index in $G$ might be infinite. Let $\rho_1$ and $\rho_2$ be two continuous representations on the finite dimensional linear space $V$ over a topological field $k$, i.e. we have continuous homomorphisms \begin{equation} \rho_i :G \rightarrow \text{GL}(V), i=1,2 \end{equation} Suppose their restrictions to $H$ is the same, $\rho_1 |_H=\rho_2 |_H$, what is the difference between the two representations?
My guess is that there exists a one dimensional representation $\phi:G \rightarrow k^\times$ whose restriction to $H$ is trivial and \begin{equation} \rho_1=\rho_2 \otimes \phi \end{equation} Since the representations of compact topological groups are determined by their characters, I tried to prove this using characters, but got stuck. Could anyone prove this statement or give a counter example?