When are all group codes of a noncommutative group Abelian (a computational approach)?

Let $G$ be a finite group and $F$ be a field. Any linear code over $F$ that is permutation equivalent to some code defined by an ideal of the group ring $FG$ will be called a $G$-code. The theory of these “abstract” group codes was developed in 2009. A code is called Abelian if it is an $A$-code for some Abelian group $A$. Some conditions were given that all $G$-codes for some group $G$ are Abelian but no examples of non-Abelian group codes were known at that time. We use a computer algebra system GAP to show that all $G$-codes over any field are Abelian if $|G|<128$ and $|G|\notin\{24,48,54,60,64,72,96,108,120\}$, but for $F=\mathbb F_5$ and $G=\mathrm S_4$ there exist non-Abelian $G$-codes over $F$. It is also shown that the existence of left non-Abelian group codes for a given group depends in general on the field of coefficients, while for (two-sided) group codes the corresponding question remains open.