Let $G$ be a group, and $$\rho:G\to \mathrm{GL}(V)$$ be an absolutely irreducible, finite-dimensional representation over a characteristic $0$ field $k$. For each finite index subgroup $H\le G$, let $$\mathrm{End}_H(V) = \{\phi\in\mathrm{End}(V) :g\cdot\phi(v) = \phi(g\cdot v)\;\;\forall v\in V, \ g\in H\}$$ be the endomorphism ring of $\rho|_H$.

Fix some finite index normal subgroup $N\lhd G$. Then there is a natural map $$\begin{align}\begin{Bmatrix}\text{Subgroups $H$ with }\\N\le H\le G\end{Bmatrix}&\to\begin{Bmatrix}\text{Subalgebras $A$ with } \\\mathrm{End}_G(V)\subseteq A\subseteq\mathrm{End}_N(V)\end{Bmatrix}\\ \\H&\mapsto \mathrm{End}_H(V).\end{align}$$

Can we determine which subalgebras $A$ are in the image of this map?

I'm particularly interested in the case where $\dim V = 4$ and $\mathrm{End}_N(V) = M_2(k)$. In this case, is there a sugbroup $H$ such that $\mathrm{End}_H(V) = k\times k$?

Edit:

My motivation is as follows. Suppose I have a representation (in my case, a Galois representation) $$\rho:G\to\mathrm{Aut}(V)\cong\mathrm{GL}_4(k)$$ and I know that for some normal subgroup $N$, $$\rho|_N = \sigma\oplus\sigma$$ for some representation $\sigma:N\to\mathrm{GL}_2(k)$. A priori, I have no information about $N$. Under what circumstances can I find an $H$ such that $$\rho|_H=\sigma_1\oplus\sigma_2,$$ where the $\sigma_i:H\to\mathrm{GL}_2(k)$ are distinct? Since I know nothing about $N$, I'm hoping for a condition that is intrinsic to the representation in some way.

Further edit:

Let $G$ act on $\mathrm{End}_N(V)$ by $$g\cdot \phi = \rho(g)\circ\phi\circ\rho(g^{-1})$$ for $g\in G$, $\phi\in \mathrm{End}_N(V)$. Enlarging $N$ if necessary, we can assume that $N$ is the kernel of this action, and we get an injection $$G/N\hookrightarrow \mathrm{Aut}(M_2(k))=\mathrm{PGL}_2(k).$$ Since $G/N$ is finite, it is either cyclic, dihedral, $A_4$, $S_4$ or $A_5$. As Johannes points out in his answer, it is only the $A_5$ case which presents a difficulty.

Are there any facts that I could use about $\rho$, without knowing what $N$ is, that would enable me to rule out this case?