Virtually abelian not implies FZ

Statement
It is possible to have a group $$G$$ that is a virtually abelian group (i.e., it has an abelian normal subgroup that has finite index) but such that $$G$$ is not a FZ-group -- the center of $$G$$ has infinite index.

Proof
The particular example::infinite dihedral group is an example. It has an abelian normal subgroup of index two -- its cyclic maximal subgroup which is an infinite cyclic group. However, it is a centerless group, so its center has infinite index.