Is it possible to derive Schur-Zassenhaus theorem (see e.g. https://en.wikipedia.org/wiki/Schur%E2%80%93Zassenhaus_theorem) from Wedderburn-Malcev theorem (see e.g. https://www.math.uni-bielefeld.de/~sek/select/RF6.pdf)?
For the existance part I have the following idea:
Let $G$ be a finite Group and $N$ an abelian Hall $p$-subgroup. Let $K$ be a field containing $p$ elements. We Focus on the natural epimorphism from $G$ onto $G/N$. This map can be enhanced to an algebra epimorphism from $KG$ onto $K(G/N)$. The kernel of this map is known: $KG\cdot Aug(KN)=Aug(KN)\cdot KG$. The Augmentation ideal $Aug(KN)$ is nilpotent based on a theorem of Wallace. Thus, $KG\cdot Aug(KN)=Aug(KN)\cdot KG$ is nilpotent, too. $N$ is a Hall subgroup, and hence $G/N$ is a $p'$-subgroup. Hence, $K(G/N)$ is semisimple by a theorem of Maschke. We deduce that $KG\cdot Aug(KN)=Aug(KN)\cdot KG$ is exactly the radical of $KG$. In addition, $K(G/N)$ is separable. By using the theorem of Wedderburn-Malcev a complement $T$ of $rad(KG)$ exists in $KG$: $KG=T\oplus rad(KG)$. We deduce that $E(KG)$ is the semidirect product of $E(T)$ and the nilpotent normal subgroup $1+rad(KG)=1+KG\cdot Aug(KN)$. $N$ is a subgroup of $1+rad(KG)$.
Is it possible to control the intersection of $E(KG)$ and $G$?