Group with finitely generated inner automorphism group

Definition
A group $$G$$ is termed a group with finitely generated inner automorphism group if the defining ingredient::inner automorphism group $$\operatorname{Inn}(G)$$ (which is isomorphic to the quotient group $$G/Z(G)$$ of $$G$$ by its center $$Z(G)$$) is a defining ingredient::finitely generated group.