Characteristic upper-hook AEP implies characteristic

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., AEP-subgroup) must also satisfy the second subgroup property (i.e., subgroup in which every subgroup characteristic in the whole group is characteristic)
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Statement

Statement with symbols

Suppose $H\le K\le G$ are groups satisfying the following two conditions:

• $H$ is a Characteristic subgroup (?) of $G$: every automorphism of $G$ restricts to an automorphism of $H$.
• $K$ is an AEP-subgroup (?) of $G$: every automorphism of $K$ can be extended to an automorphism of $G$.

Then, $H$ is a characteristic subgroup of $K$.

Related facts

Similar facts

• Normal upper-hook fully normalized implies characteristic
• Fully invariant upper-hook EEP implies fully invariant
• AEP upper-hook characteristic implies AEP

Proof

Given': Groups $H\le K\le G$ such that $H$ is characteristic in $G$ and $K$ is an AEP-subgroup of $G$.

To prove: If $\sigma$ is an automorphism of $K$, $\sigma (H)=H$.

Proof: Let $\sigma$ be an automorphism of $K$. Since $K$ is AEP in $G$, $\sigma$ extends to an automorphism ${\sigma }^{\prime }$ of $G$. ${\sigma }^{\prime }(H)=H$ since $H$ is characteristic in $G$. Thus, $\sigma (H)=H$.