Assume $H$ is trivial, and $\rho$ a sum of two irreducible, non-isomorphic representations $V=V_1+V_2$ then $End_G(V) = \mathbb{C}^2$ and you have $End_k(V) = End_H(V)$. Because $\rho(G)$ will preserve the $V_i$'s, your conjecture will fail in that case. You can't construct an operator mapping $V_1 \rightarrow V_2$ with elements of type $\rho(g)$.
If $\rho$ is irreducible and $H$ is trivial, you claim that $\rho(G)$ generates $End_k(V)$. You have that $\{ \rho(g) v : g \in G\}$ spans $V$, but I would guess that isn't sufficient as well.