Center not is surjective endomorphism-balanced

Statement with symbols
It is possible to have a group $$G$$ and a surjective endomorphism $$\sigma$$ of $$G$$ such that the restriction of $$\sigma$$ to the center $$Z(G)$$ is not surjective as an endomorphism of $$Z(G)$$.

Other proofs of similar subgroup-defining functions not being surjective endomorphism-balanced

 * Baer norm not is surjective endomorphism-balanced
 * Wielandt subgroup not is surjective endomorphism-balanced

Proofs of similar subgroup-defining functions being strictly characteristic

 * Center is strictly characteristic
 * Baer norm is strictly characteristic
 * Wielandt subgroup is strictly characteristic

Facts used

 * 1) uses::Isomorphic to inner automorphism group not implies centerless: We can have a group $$G$$ such that the center $$Z(G)$$ is nontrivial, but there is an isomorphism between $$G/Z(G)$$ and $$G$$.

Proof using fact (1)
Let $$G$$ be an example group for fact (1). Let $$\alpha:G \to G/Z(G)$$ and the quotient map and $$\varphi:G/Z(G) \to G$$ be an isomorphism. Then, $$\sigma = \varphi \circ \alpha$$ is a surjective endomorphism of $$G$$. On the other hand, the restriction of $$\sigma$$ to $$Z(G)$$ is the trivial map, which is not surjective by the assumption that $$Z(G)$$ is a nontrivial group.