Center not is intermediately characteristic

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

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 $$\{ e,a^2,x,a^2x \}$$.