AEP upper-hook characteristic implies AEP

Statement

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

• $H$ is an AEP-subgroup (?) of $G$, i.e., every automorphism of $H$ extends to an automorphism of $G$.
• $K$ is a 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