Center not is endomorphism kernel

This article gives the statement, and possibly proof, of the fact that for a group, the subgroup obtained by applying a given subgroup-defining function (i.e., center) does not always satisfy a particular subgroup property (i.e., endomorphism kernel)
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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

Further information: subgroup structure of quaternion group

The simplest example is where $G$ is the quaternion group, so $Z(G)$ is the 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.