Center not is fully invariant

Statement
The center of a group need not be a fully invariant subgroup.

Related facts

 * Center not is fully invariant in class two p-group
 * Center not is normality-preserving endomorphism-invariant: We can use the same idea, but need to impose some more constraints on the generic example.

Generic example
Let $$A$$ be an abelian group and $$C$$ a centerless group containing a subgroup isomorphic to $$A$$, say $$B$$. Consider the direct product $$A \times C$$.

Clearly, $$A$$ is the center of $$A \times C$$.

Now consider the endomorphism of $$A \times C$$ which composes the projection onto $$A$$ with the isomorphism from $$A$$ to $$B$$. This endomorphism does not send $$A$$ to within itself, and hence, $$A$$ is not a fully characteristic subgroup of $$C$$.

Thus, the center is not fully invariant.

Particular example
The particular example can be obtained from the generic example above by setting $$A$$ to be a cyclic group of order 2 and $$B = S_3$$ (the symmetric group on three letters). $$B$$ in this case is the 2-element subgroup generated by a transposition.