Serial group rings of finite groups. $p$-nilpotency

We prove that for every finite $p$-nilpotent group $G$ with a cyclic $p$-Sylow subgroup and any field of characteristic $p$, the group ring $FG$ is serial. As a corollary we show that the group ring of a finite group oven an arbitrary field of characteristic $2$ is serial if and only if its $2$-Sylow subgroup is cyclic.