Center not is image-closed characteristic

Statement
The center of a group need not be an image-closed characteristic subgroup, i.e., its image under a surjective homomorphism need not be a characteristic subgroup of the image.

Generic example
Let $$A$$ be an Abelian group and $$G$$ a centerless group such that $$A$$ is isomorphic to the quotient of $$G$$ by some normal subgroup $$N$$. Consider the direct product $$A \times G$$.

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

Now consider the quotient map by $$N$$. Under this quotient map, the group $$A \times G$$ maps to $$A \times G/N$$, with $$A$$ mapping to the direct factor $$A$$ in $$A \times G/N$$. By our assumption, $$G/N$$ is isomorphic to $$A$$, so this quotient group looks like $$A \times A$$. Clearly, $$A$$ is not characteristic in this.

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 $$G = S_3$$ (the symmetric group on three letters). $$N$$ in this case is the subgroup of order three generated by a 3-cycle.