Examples of a group $G$ with a non-trivial homomorphism $f:G \to Z(G)$

Question

I recently learned that if $f: G \to Z(G)$ is a homomorphism of $G$ to its center, then $g:G \to G$ defined as $g(x)=f(x)x$ is an endomorphism of $G$.

I am having trouble thinking of examples of a (finite) group with a non-trivial homomorphism from itself to its center. This excludes trivial centers and abelian groups. Can someone give me at least two examples?

EDIT: It occurs to me that if $G=H \times A$ for some group $H$ with non-trivial center and abelian group $A$, then the map $f:(x,y) = y$ is a homomorphism. In essence, the automorphism $g$ is then $g(x,y) = (x,y^2)$. An example where $G$ cannot be decomposed as such would be appreciated.

EDIT: I think I have some confusion about the properties of $f$ in order that $g$ must be an automorphism. That is, I am unsure if the image of $f$ must be the entire center or not.

Answer 1

For any finite field $F$ and any $n$, you can consider $G:=\operatorname{GL}_n$, whose center is $F^\times I$ where $I$ is the identity matrix. Now, take the determinant as your homomorphism.

Answer 2

Consider $G = D_{n} = \langle r, s | r^n, s^2, srsr \rangle$, for $n = 2k$ even. Then $H = \langle r \rangle$ is normal in $G$, and $$G/H \cong C_2 \cong Z(G) = \langle r^k \rangle.$$ So composing the projection $G \to G/H$ with this isomorphism gives you a map of the desired kind, and I think it's non-trivial in all the ways you've requested. I should also take $k$ to be even for the reason described in the comments below.