Center not is weakly normal-homomorph-containing

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., weakly normal-homomorph-containing subgroup)
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Statement

It is possible to have a group $G$ with center $Z$ and a homomorphism $f:Z\to G$ such that $f$ sends normal subgroups of $G$ contained in $Z$ to normal subgroups of $G$ but $f(Z)$ is not contained in $Z$.

Facts used

1. Center not is normality-preserving endomorphism-invariant

Proof

The proof uses Fact (1). In particular, the same generic example works, as does the same particular example of direct product of S3 and Z3.