Characteristic direct factor not implies fully invariant

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

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.