Normal not implies potentially fully invariant

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal subgroup) need not satisfy the second subgroup property (i.e., potentially fully invariant subgroup)
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Statement

It is possible to have a normal subgroup $H$ of a group $G$ that is not a potentially fully invariant subgroup of $G$ -- in other words, there is no group $K$ containing $G$ such that $H$ is a fully invariant subgroup of $K$ .

Related facts

Normal not implies potentially verbal
NPC theorem: Normal equals potentially characteristic.
Normal equals potentially normal-subhomomorph-containing

Proof

Let $A$ be a nontrivial complete group. Define $G : = A \times A$ and $H : = A \times {e}$ . Clearly, $H$ is a normal subgroup of $G$ .

Suppose $K$ is a group containing $G$ , such that $H$ is fully invariant in $K$ . In particular, $H$ is normal in $K$ . Since $H$ is complete, it is a direct factor, so there exists a group $C$ that is a complement to $H$ , so $K = H \times C$ as an internal direct product. Further, since $G / H ≅ H ≅ A$ is a subgroup of $K / H$ , $C$ has a subgroup, say $B$ , isomorphic to $A ≅ H$ .

Then, consider the endomorphism $α$ of $K$ that sends $C$ to the trivial subgroup and $H$ isomorphically to the subgroup $B$ does not send $H$ to within itself.