Characteristic direct factor not implies 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., characteristic direct factor) need not satisfy the second subgroup property (i.e., fully invariant subgroup)
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Statement

A characteristic direct factor of a group (i.e., a characteristic subgroup that is also a Direct factor (?)) need not be a fully invariant subgroup -- it need not be invariant under all endomorphisms of the group.

Proof

A generic example

Consider $G$ to be a nontrivial centerless group, and $A$ be isomorphic to a nontrivial Abelian subgroup $B$ of $G$. Then, consider the group $K=G\times A$ and the subgroup $A_{1}=\{e\}\times A$.

• $A_{1}$ is a direct factor of $K$: By construction.
• $A_{1}$ is characteristic in $K$: In fact, $A_{1}$ equals the center of the direct product.
• $A_{1}$ is not fully invariant in $K$: We can construct an endomorphism whose kernel is the direct factor $G$, and which maps the direct factor $A$ (and hence the subgroup $A_{1}$) to the subgroup $B$.

The same generic example also shows that the center need not be a fully characteristic subgroup.