Center not is endomorphism kernel

Statement
It is possible to have a group $$G$$ such that the center $$Z(G)$$ is not an endomorphism kernel in $$G$$, i.e., $$G$$ does not have any subgroup isomorphic to its inner automorphism group.

Proof
The simplest example is where $$G$$ is the particular example::quaternion group, so $$Z(G)$$ is the particular example::center of quaternion group and the quotient group is isomorphic to the Klein four-group. $$G$$ has no subgroup isomorphic to the Klein four-group.