The characteristic subgroup of order $m$ (odd) in a group of order $2^nm$ with cyclic 2-sylow

Question

The question comes from the top answer from Prove that there is no element of order $8$ in $SL(2,3)$

It says that "the claim follows by induction on $r$ (since a normal subgroup of order $m$ is a Hall subgroup and thus characteristic)."

What I try to continue through one of the following:

1. Say at the end of induction chain we jump from $N \to M$, where $N$ has order $2m$ and $M$ has order $m$ and contain all even elements from $N$. I want to say that automorphism preserves parity, but doesn't that $S_6$ exception prevent us from doing so?
2. I understand Hall subgroup's definition but do not know how to make a jump to the characteristic property?

Answer 1

A Hall subgroup of a finite group $G$ is a subgroup $H$ of order $m$ such that $m$ and $|G|/m$ are coprime.

Suppose that $H$ is a normal Hall subgroup of $G$ of order $m$. We claim that $H$ is the unique subgroup of order $m$ in $G$, which implies that it is characteristic.

To see this, suppose that $K$ is another subgroup with $|K|=m$. Since $H$ is normal in $G$, $HK$ is a subgroup of $G$ of order $|H||K|/|H \cap K| = m|K:H \cap K|$.

If $H \ne K$, then $|K:M \cap K| \ne 1$, and it divides $|K| = |H|$, but it also divides $|G:H|$, contradicting the coprimality of $m$ and $|G:H|$. So $H=K$ and $H$ is the unique subgroup of order $m$.