Is this condition equivalent to being indecomposable for a representation? Suppose that given a finite group $G$, and given a finite dimensional vector space $V$ on a field $\mathbb{K}$ (I am thinking about fields with characteristic $p$ here, so I can't use Maschke's theorem) with a representation $\rho:G \to \operatorname{GL}(V)$, there is a subspace $W \subseteq V$ such that $W$ is stable (that is, for all $g \in G$, $\rho(g)(W) \subseteq W$).
Also, suppose that any complement $Z$ of $W$ in $V$ (that is, a subspace such that $W \oplus Z = V$) is not $G$-stable. 
Moreover, suppose that the representation $W$ obtained by restriction and the representation $V/W$ obtained by composing with the quotient map are both indecomposable (or even irreducible).
Does this imply that $V$ is an indecomposable representation?
 A: This is not enough for $V$ to be indecomposable.  It would be if your condition were true for all subrepresentations $W$.
Here is a counter-example.  Let $k=\mathbb{F}_3$ be the field with $3$ elements, and let $G=C_3$ be the cyclic group of order $3$.  Let $P$ be the representation of $C_3$ with underlying vector space $k^3$ and with an action of $C_3$ by cyclic permutations on a basis of $k^3$ (as a $kC_3$-module, $P$ is isomorphic to $kC_3$).
Then $P$ is indecomposable, and has a one-dimensional subrepresentation $$P_1 = \{(x,y,z)\in k^3 \ | \ x=y=z \}$$ and a two-dimensional subrepresentation $$P_2=\{(x,y,z)\in k^3 \ | \ x+y+z=0 \}$$
such that $P_1\subset P_2\subset P_3$.  Note that $P_1$ and $P_2$ are both indecomposable as well.  Finally, define the morphisms
$$  \pi_2:P\to P_2: (x,y,z) \to (x-y,y-z,-x+z) \quad \textrm{(surjective, with kernel $P_1$)} $$
$$\pi_{2,1}:P_2\to P_1:(x,y,z)\to (x-y,y-z,-x+z)  \quad \textrm{(surjective, with kernel $P_1$)}$$
$$\iota_2:P_2\to P \quad \textrm{(inclusion)}$$
$$\iota_{1,2}:P_1\to P_2 \quad \textrm{(inclusion)}.$$
Then there is a short exact sequence
$$ 0 \to P_2 \stackrel{(\iota_2, \pi_{2,1})^t}{\longrightarrow}  P \oplus P_1 \stackrel{(\pi_2, -\iota_{1,2})}{\longrightarrow} P_2 \to 0.$$
Take $V=P\oplus P_1$ and $W$ to be the image of the first morphism of the above short exact sequence.  Then $V$ is not indecomposable, even though both $W$ and $V/W$ are, and $W$ does not admit any $C_3$-stable complement in $V$.
