Maschke's theorem and possible generalization?

Question

One way to state maschke's theorem is that for a finite group $G$ and a field $k$ the following holds. If we see $k[G]$ as a left-module over itsself then $k[G]$ is semi-simple if and only if $\text{char}(k) \nmid |G|$. Now we want to generalize it as follows. Let $G$ act transitively on a fnite set $X$ and let $F(X)$ be the free vector space of $X$ over $k$. Then we can see $F(X)$ as a left-module over $k[G]$ via the group action. It is easy to show that if $F(X)$ is semi-simple then $\text{char}(k)\nmid |X|$. However i am not sure wether the converse holds or not. Does someone know a counterexample or a proof?

Answer 1

The converse is false in general. I found a counterexample after a few tries with a computer calculation.

Let $G=A_5$ acting on transitively on $15$ points, Then the permutation module over ${\mathbb F}_4$ (a splitting field in characteristic $2$) has two indecomposable components of dimension $5$ with composition factors of dimensions $2$, $1$, and $2$.