In this question, all representations are finite-dimensional representations over $\mathbb{C}$.

Fix some $n \geq 3$. Assume that $V$ is a representation of $\text{SL}(n,\mathbb{Z})$. Also, assume that $W$ is a subrepresentation of $V$ and set $V' = V/W$, so we have a short exact sequence

$$0 \longrightarrow W \longrightarrow V \longrightarrow V' \longrightarrow 0$$

of $\text{SL}(n,\mathbb{Z})$ representations. Assume that the actions of $\text{SL}(n,\mathbb{Z})$ on $W$ and $V'$ extend to actions of $\text{SL}(n,\mathbb{C})$. Question : Must the action of $\text{SL}(n,\mathbb{Z})$ on $V$ extend to an action of $\text{SL}(n,\mathbb{C})$?

Margulis superrigidity says that the action virtually extends, but I can't construct representations where it doesn't extend on the nose. Of course, if it extends then $V = V' \oplus W$ as $\text{SL}(n,\mathbb{C})$-representations.