Series-equivalent characteristic subgroups may be distinct

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., series-isomorph-free subgroup)
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Statement

It is possible to have a group $G$, a characteristic subgroups $H$ and $K$ of $G$ such that $H$ and $K$ are isomorphic groups and the quotient groups $G/H$ and $G/K$ are isomorphic groups. In other words, there can be distinct characteristic subgroups that are Series-equivalent subgroups (?).

In particular, a characteristic subgroup of $G$ need not be a series-isomorph-free subgroup of $G$.

Related facts

Stronger facts

There are two slightly stronger facts that are true, either of which can be used to supply examples:

• Series-equivalent characteristic central subgroups may be distinct
• Characteristic maximal subgroups may be isomorphic and distinct in group of prime power order

Proof

See the examples in the proof of either of the stronger facts.