Let $G$ be a finite group and let $k$ be a finite field with char$(k)=p$ such that $p\mid |G|$.

If $k$ is a splitting field for $G$, then, no matter which splitting field we take, after extending scalars we always have the same number of isomorphism classes of simple modules.

However, it is not necessarily true that every indecomposable $kG$-module is absolutely indecomposable.

This leads to the following

Question: Is there a general construction that gives us a finite splitting field $K$ for $G$ (and all subgroups of all factor groups of $G$) such that every indecomposable $KG$-module (and every indecomposable K[H/J]-module where H/J is a factor group of a subgroup of G) is absolutely indecomposable?

Is there a reference in the literature?

Thanks for the help.