Let $G$ be finite group. Let $k$ be a finite field and $K$ an algebraically closed field containing $k$.
An irreducible $k[G]$-module $V$ is said to be absolutely irreducible if $V \otimes_k K$ is an irreducible $K[G]$-module.
Suppose that $V$ is a $k[G]$-module such that every composition factor of $V$ is absolutely irreducible. Do $V$ and $V \otimes_k K$ have the same submodule structure?
That is, is it true that every $K[G]$-submodule of $V \otimes_k K$ is of the form $W \otimes_k K$, where $W$ is a $k[G]$-submodule of $V$?
Suppose that V is trivial of dimension 2. Then V has a finite number of submodules but the extended module has infinitely many submodules.