Series-equivalent not implies automorphic
It is possible to have a group and normal subgroups and of that are Series-equivalent subgroups (?) in the sense that and , but and are not automorphic subgroups -- in other words, there is no automorphism of that sends to .
There are many slight strengthenings of the result that are presented below, along with the smallest order of known examples.
For the proof, see any of the stronger facts listed above.