Center not is intermediately characteristic

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., intermediately characteristic subgroup)
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Statement

The center of a group need not be characteristic in every intermediate subgroup.

Proof

Example of the dihedral group

In the dihedral group of order eight, generated by a rotation $a$ and reflection $x$, the center $\{e,a^{2}\}$ is not intermediately characteristic. In fact, it is not characteristic in the intermediate subgroup $\{e,a^{2},x,a^{2}x\}$.