Let $G$ be a finite group, $p$ a prime number. We denote by $\mathbb{F}_p$ the field of cardinality $p$. Let $V$ be an infinite dimensional representation of $G$ over $\mathbb{F}_p$.

Must there be $G$-invariant, proper subspaces $U,W \leq V$ such that $U + W = V$?

I do not require the sum to be direct. The question should be equivalent to asking:

Must $V$ have a nontrivial decomposable image?

I am able to prove this if $p$ does not divide $|G|$ by applying Maschke's theorem to decompose $V$ into a direct sum of finite-dimensional irreducible subrepresentations. Even if in general the answer is negative, I would like to know about additional cases in which the conclusion holds, to say:

Under what conditions on $G$ can we find such subspaces?

Interesting cases can be abelian, solvable or any other "nice" classes of groups.