Automorph-conjugacy is transitive

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 $$g \sigma(H) g^{-1} = hHh^{-1}$$. Rearranging, we see that $$\sigma(H) = g^{-1}h H h^{-1}g$$, a conjugate fo $$H$$.