Series-equivalent characteristic central subgroups may be distinct

Statement
It is possible to have a finite group $$G$$ (in fact, even a group of prime power order) and characteristic subgroups $$H$$ and $$K$$ of $$G$$ such that:


 * 1) $$H$$ and $$K$$ are fact about::series-equivalent subgroups of $$G$$: $$H$$ and $$K$$ are isomorphic groups and the quotient groups $$G/H$$ and $$G/K$$ are also isomorphic groups.
 * 2) $$H$$ and $$K$$ are both fact about::central subgroups of $$G$$, i.e., they are both contained in the center of $$G$$. In particular, both are fact about::characteristic central subgroups of $$G$$.
 * 3) $$H$$ is not equal to $$K$$, i.e., $$H$$ and $$K$$ are distinct subgroups.

Related facts
See series-equivalent not implies automorphic.

Proof
The smallest example is where $$G$$ is SmallGroup(32,28), $$H$$ and $$K$$ are both characteristic central subgroups of order 2, and $$G/H$$ and $$G/K$$ are both isomorphic to direct product of D8 and Z2.