AEP upper-hook characteristic implies AEP

Statement
Suppose $$H \le K \le G$$ are groups, such that the following two conditions hold:


 * $$H$$ is an fact about::AEP-subgroup of $$G$$, i.e., every automorphism of $$H$$ extends to an automorphism of $$G$$.
 * $$K$$ is a fact about::characteristic subgroup of $$G$$, i.e., every automorphism of $$G$$ restricts to an automorphism of $$K$$.

Then, $$H$$ is an AEP-subgroup of $$K$$.

Related facts

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