Characteristic upper-hook AEP implies characteristic

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


 * $$H$$ is a fact about::characteristic subgroup of $$G$$: every automorphism of $$G$$ restricts to an automorphism of $$H$$.
 * $$K$$ is an fact about::AEP-subgroup of $$G$$: every automorphism of $$K$$ can be extended to an automorphism of $$G$$.

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

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 $$\sigma'$$ of $$G$$. $$\sigma'(H) = H$$ since $$H$$ is characteristic in $$G$$. Thus, $$\sigma(H) = H$$.