Center not is endomorphism kernel

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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.