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 fact about::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 $$G$$.

Applications

 * Isomorph-normal coprime automorphism-invariant of Sylow implies weakly closed

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 $$K$$.