Intermediate characteristicity is not transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., intermediately characteristic subgroup) not satisfying a subgroup metaproperty (i.e., transitive subgroup property).
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Statement

An intermediately characteristic subgroup of an intermediately characteristic subgroup need not be intermediately characteristic.

Proof

Example of the dihedral group

Further information: Dihedral group:D8, Subgroup structure of dihedral group:D8

Consider the dihedral group:D8, the dihedral group acting on a set of size four, i.e., the dihedral group with eight elements, given explicitly by the presentation:

$\langle a,x \mid a^4 = x^2 = e, xax^{-1} = a^{-1} \rangle$.

The cyclic subgroup generated by $a$ is a subgroup of order four, and since all the elements outside it have order two, it is a characteristic subgroup. Being maximal, it is intermediately characteristic. Within this, the cyclic subgroup of order two generated by $a^2$ is again intermediately characteristic.

However, the cyclic subgroup of order two generated by $a^2$ is not intermediately characteristic in the whole group: it is isomorphic to the two-element subgroup generated by $x$.