Coprime automorphism-invariant normal subgroup of Hall subgroup is normalizer-relatively normal

Definition

Suppose $G$ is a finite group, $K$ is a Hall subgroup of $G$, and $H$ is a Coprime automorphism-invariant normal subgroup (?) of $G$. Then, $H$ is normal in the normalizer $N_G(K)$. In other words, $H$ is normalizer-relatively normal in $K$ relative to $G$.

Proof

Given: A finite group $G$, a Hall subgroup $K$ of $G$, a coprime automorphism-invariant normal subgroup $H$ of $K$.

To prove: $H$ is normal in $N_G(K)$.

Proof: Let $g \in N_G(K)$. Suppose $g$ has order $m$.

1. We can write $g = ab$ where $a \in K$ and the order of $b$ is relatively prime to the order of $K$.
2. Conjugation by $g$ can be expressed as a composite of conjugation by $a$ and conjugation by $b$, and $H$ is invariant under both: $H$ is invariant under conjugation by $b$ because this is an automorphism of coprime order. $H$ is invariant under conjugation by $a$ because $H$ is normal in $K$ and $a$ is in $K$.