The well known Maschke Theorem in group representation theory says: Let $G$ be a finite group and $F$ be a field so that $charF\nmid |G|$. Then every $F[G]$-module $V$ is completely reducible. That is for every submodule $W$ of $V$, we have $V=W\oplus U$ where $U$ is also a submodule of $V$. I am wondering what will happen if I change submodule $W$ by a subspace of $V$, say $X$. Will $X$ have a submodule as a complement in $V$? Generally, it is not true.
Now let $K$ be a field of order $p$, $G$ be a finite group and $(|G|,p)=1$, $V$ be a faithful homogenous $K[G]$-module so that $V=U\oplus U$ where $U$ is an irreducible $K[G]$-module. Apparently, $U$ is also a faithful $K[G]$-module. Now let $X$ be any subspace of $V$ with $\dim_K(X)=\dim_K(U)$, and I am wondering in what circumstances there exists a submodule $W$ such that $V=X\oplus W$.