Fully invariant upper-hook EEP implies fully invariant

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


 * $$H$$ is a fact about::fully invariant subgroup of $$G$$: every endomorphism of $$G$$ restricts to an endomorphism of $$H$$.
 * $$K$$ is an fact about::EEP-subgroup of $$G$$: every endomorphism of $$G$$ restricts to an endomorphism of $$K$$.

Then, $$H$$ is a fully invariant subgroup of $$K$$.

Related facts

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

Proof
Given: $$H \le K \le G$$, $$H$$ is fully invariant in $$G$$ and $$K$$ is EEP in $$G$$.

To prove: For any endomorphism $$\alpha$$ of $$K$$, $$\alpha(H) \le H$$.

Proof: Since $$K$$ is EEP in $$G$$, there exists an endomorphism $$\alpha'$$ of $$G$$ such that the restriction of $$\alpha'$$ to $$K$$ equals $$\alpha$$. Since $$H$$ is fully invariant in $$G$$, $$\alpha'(H) \le H$$. Thus, $$\alpha(H) \le H$$.