Series-equivalent characteristic central subgroups may be distinct

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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 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 Central subgroup (?)s of G, i.e., they are both contained in the center of G. In particular, both are Characteristic central subgroup (?)s 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#Related facts.

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.