Projector additive group codes

Let $F=\mathbb{F}_q$ and let $K=\mathbb{F}_{q^m}$ be a finite extension of $F$. An additive left group code is a left $FG$-submodule of the group algebra $KG$. Classical idempotent group codes describe images of $KG$-linear projectors and are necessarily $K$-linear. They therefore do not provide a s