Is a subnormal Hall subgroup of a finite group normal?

Question

Recall that a subgroup $H$ of a finite group $G$ is called Hall subgroup if $(|H|,[G:H])=1$.

If $H$ is a subnormal Hall subgroup of finite group $G$, then is $H$ normal?

Answer 1

Yes this is true: if $H=H_0 \lhd H_1 \cdots \lhd H_i \lhd \cdots \lhd G=H_r$ is a subnormal series then $H$ is a normal Hall subgroup in $H_1$. And hence it is characteristic in $H_1$ (every automorphism of the group $H_1$ takes $H$ to itself). So $H$ char $H_1 \lhd H_2$, which implies that $H$ is normal in $H_2$. Now apply induction on $i$.