Center not is image-closed characteristic

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., image-closed characteristic subgroup)
View subgroup property satisfactions for subgroup-defining functions $|$ View subgroup property dissatisfactions for subgroup-defining functions

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.

Proof

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.