Series-equivalent not implies automorphic

Statement

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 .

Related facts

Stronger facts

There are many slight strengthenings of the result that are presented below, along with the smallest order of known examples.

Statement	Constraint on	Smallest order of among known examples	Isomorphism class of	Isomorphism class of	Isomorphism class of quotient group
series-equivalent abelian-quotient abelian not implies automorphic	and are both abelian	16	nontrivial semidirect product of Z4 and Z4	direct product of Z4 and Z2	cyclic group:Z2
series-equivalent characteristic central subgroups may be distinct	and are both central subgroups of	32	SmallGroup(32,28)	cyclic group:Z2	direct product of D8 and Z2
series-equivalent abelian-quotient central subgroups not implies automorphic	and are central and are abelian	64	semidirect product of Z8 and Z8 of M-type	direct product of Z4 and Z2	direct product of Z4 and Z2
series-equivalent not implies automorphic in finite abelian group	is a finite abelian group	128	direct product of Z8 and Z4 and V4	direct product of Z8 and V4	direct product of Z4 and Z2
characteristic maximal not implies isomorph-free in group of prime power order	and are maximal, is characteristic, and is a group of prime power order	16	nontrivial semidirect product of Z4 and Z4	direct product of Z4 and Z2	cyclic group:Z2
characteristic maximal subgroups may be isomorphic and distinct in group of prime power order	Both and are characteristic and maximal and is a group of prime power order	64	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.	PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] One of the people editing this page intended to fill in this information at a later stage, but hasn't gotten around to doing it yet. If you see this placeholder for a long time, file an error report at the error reporting page.

Proof

For the proof, see any of the stronger facts listed above.