Automorph-conjugacy is transitive

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Statement

Property-theoretic statement

The subgroup property of being automorph-conjugate is transitive.

Symbolic statement

Let $H$ be an automorph-conjugate subgroup of $K$, and $K$ be an automorph-conjugate subgroup of $G$. Then, $H$ is an automorph-conjugate subgroup of $G$.

Proof

Suppose $H$ is an automorph-conjugate subgroup of $K$, and $K$ is an automorph-conjugate subgroup of $G$. We want to show that $H$ is an automorph-conjugate subgroup of $G$.

For this, pick any automorphism $\sigma$ of $H$. Clearly, $\sigma (H)\le \sigma (K)$, and since $K$ is automorph-conjugate subgroup of $G$, there exists $g\in G$ such that $g\sigma (K)g^{-1}=K$. Thus, $c_g\circ \sigma$ (conjugation by $g$, composed with $\sigma$), gives an automorphism of $K$. Since $H$ is automorph-conjugate inside $K$, there exists $h\in K$ such that $g\sigma (H)g^{-1}=hH{h}^{-1}$. Rearranging, we see that $\sigma (H)=g^{-1}hH{h}^{-1}g$, a conjugate fo $H$.