Let $G$ be a group. Let $(V,\tau)$ be a representation of $G$ where $V$ is a finite dimensional vector space over $\mathbb{C}$. $\tau: G \longrightarrow GL(V)$ is a group homomorphism. Then we can identify each element of $G$ as a linear map $T \longrightarrow T$. Let $W \subset V$, $W$ is a subrepresentation if for all $g \in G, w \in W$ we have $wg \in W$.
A representation homomorphism: $\theta: V_1 \longrightarrow V_2$ is a linear map such that $\theta(v_1g) = \theta(v_1)g$.
Here is my question, Let $V$ be a representation of $G$. Let $W \subset V$ be a subrepresentation. Then obviously there exists an injective homomorphism $\alpha: W \longrightarrow V$.
But does there exist $\beta: V \longrightarrow W$ such that $\beta$ is a surjective homomorphism between representations?
If $G$ is finite, then we can simply use Maschke's theorem, then take the projection map as $\beta$. But what if $G$ is infinite, my intuition is that $\beta$ might not exist.
I tried to play around with the example given here example of infinite group that maschke's theorem is not hold.
I know that the projection map does not exist, but how about just surjection?