ACIC is characteristic subgroup-closed

Statement
Suppose $$G$$ is an ACIC-group, i.e., every automorph-conjugate subgroup of $$G$$ is characteristic in $$G$$ (note that characteristic implies automorph-conjugate, so in such a group, the notion of being a characteristic subgroup precisely coincides with the notion of being an automorph-conjugate subgroup).

Suppose $$H$$ is a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) of $$G$$. Then, $$H$$ is also an ACIC-group.

Generalizations

 * Transitive implies intermediate subgroup condition implies closed under former

Facts used

 * 1) uses::Automorph-conjugacy is transitive
 * 2) uses::Normality satisfies intermediate subgroup condition

Proof
Given: An ACIC-group $$G$$, a characteristic subgroup (or equivalently, an automorph-conjugate subgroup) $$H$$ of $$G$$, and an automorph-conjugate subgroup $$K$$ of $$H$$.

To prove: $$K$$ is normal in $$H$$, or equivalently, $$K$$ is characteristic in $$H$$.

Proof: