Center not is weakly normal-homomorph-containing

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) uses::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 particular example::direct product of S3 and Z3.