$G$ is a finite group with a subgroup $H$. Let $\rho_1:G \to GL(V)$ and $\rho_2:H \to GL(U)$ be irreducible representations. $Z=\mathbb{C}[G]^H$, i.e., $Z$ is the centralizer of $H$ in $\mathbb{C}[G]$.
Show that the following defines an action of $Z$ on the space $\def\Hom{\operatorname{Hom}}\Hom_H(U,V )$ of functions: $f\mapsto g.f := \rho_1(g)\circ f$. Further, Show that this action is a transitive action (or equivalently $\Hom_H(U,V)$ is an irreducible $C[G]^H$ module).
It's easy to see that the given map is an action of $Z$ on $\Hom_H(U,V)$. But I am unable to prove that the action is transitive i.e. if $f_1,f_2$ (nonzero) belongs to $\Hom_H(U,V) $ then there exists $g$ in $Z$ such that $g.f_1=f_2$. Any Hints/ideas ?