Automorph-permutable of normal implies conjugate-permutable

Property-theoretic statement
Automorph-permutable * Normal $$\le$$ Normal

here $$*$$ denotes the composition operator.

Verbal statement
Any automorph-permutable subgroup of a normal subgroup is conjugate-permutable.

Symbolic statement
Suppose $$H \le K \le G$$ are groups, such that $$H$$ is automorph-permutable in $$K$$, and $$K$$ is normal in $$G$$. Then $$H$$ is conjugate-permutable in $$G$$.

Related facts

 * Characteristic of normal implies normal: Both proofs use very similar arguments.
 * 2-subnormal implies conjugate-permutable: This statement follows from the statement of this article, using the fact that normal implies automorph-permutable.

Proof
Suppose $$H \le K \le G$$ are groups, such that $$H$$ is automorph-permutable in $$K$$, and $$K$$ is normal in $$G$$. We need to show that $$H$$ is conjugate-permutable in $$G$$.

We do this as follows. First, pick $$g \in G$$, and consider the operation $$c_g:x \mapsto gxg^{-1}$$. We need to show that $$H$$ and $$c_g(H) = H^g = gHg^{-1}$$ permute. Since $$K$$ is a normal subgroup of $$G$$, $$c_g$$ restricts to an automorphism of $$K$$, and thus $$c_g(H)$$ (simply $$gHg^{-1}$$ is an automorph of $$H$$ in $$K$$). Since $$H$$ is automorph-permutable inside $$K$$, $$H$$ permutes with $$c_g(H)$$, completing the proof.