I was presented with this question as a study problem in my representation theory course in college and have spent hours trying to solve it or find something similar. Any help or direction on this problem would be greatly appreciated
Let $H$ be a normal subgroup of $G$ ($gHg^{-1} =H$ for all $g \in G$). Let $P^h_v: G → GL(V)$ be a representation of $G$. Let $\rho^H_v: H \to G \to GL(V)$ be the restriction of the representation to $H$. Let $V = \displaystyle\bigoplus_{i=1}^n W_i$ be the isotypical decomposition of $V$. Prove that for any $g \in G$ and $i \in \{1,\cdots,n\}$ there is a $j \in \{1,\cdots,n\}$ such that $g(W_i) = W_j$ and $(g\cdot h)(W_i) = W_j$ for any $h \in H$
Hint: show that if $U$ is an irreducible $H$ module and $g \in G$, then $gU$ is also an irreducible $H$ module and if $U_1 \cong U_2$ are isomorphic as $H$ modules, then $gU_1 \cong gU_2$ are also isomorphic as $H$ modules. Use this to show that isotypic components must go to isotypic components as the problem asks.