Marginality does not satisfy intermediate subgroup condition

This article gives the statement, and possibly proof, of a subgroup property (i.e., marginal subgroup) not satisfying a subgroup metaproperty (i.e., intermediate subgroup condition).
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Statement

It is possible to have a group $G$ and subgroups $H,K$ of $G$ such that $H\leq K$ and:

• $H$ is a marginal subgroup of $G$.
• $H$ is not a marginal subgroup of $K$.

Proof

Further information: dihedral group:D8, subgroup structure of dihedral group:D8

Consider the following:

• $G$ is dihedral group:D8.
• $H$ is the center of dihedral group:D8.
• $K$ is one of the Klein four-subgroups of dihedral group:D8.

Then, we have:

• $H$ is marginal in $G$, because center is marginal (it is marginal with respect to the variety of abelian groups).
• $H$ is not marginal in $K$, because it is not characteristic in $K$.