Series-equivalent characteristic subgroups may be distinct

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 fact about::series-equivalent subgroups.

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

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


 * Weaker than::Series-equivalent characteristic central subgroups may be distinct
 * Weaker than::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.