Series-equivalent not implies automorphic

Statement

It is possible to have a group $G$ and normal subgroups $H$ and $K$ of $G$ that are Series-equivalent subgroups (?) in the sense that $H \cong K$ and $G/H \cong G/K$, but $H$ and $K$ are not automorphic subgroups -- in other words, there is no automorphism of $G$ that sends $H$ to $K$.

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 $G,H,K$	Smallest order of $G$ among known examples	Isomorphism class of $G$	Isomorphism class of $H,K$	Isomorphism class of quotient group $G/H, G/K$
series-equivalent abelian-quotient abelian not implies automorphic	$H$ and $G/H$ 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	$H$ and $K$ are both central subgroups of $G$	32	SmallGroup(32,28)	cyclic group:Z2	direct product of D8 and Z2
series-equivalent abelian-quotient central subgroups not implies automorphic	$H$ and $K$ are central and $G/H, G/K$ 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	$G$ 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	$H$ and $K$ are maximal, $H$ is characteristic, and $G$ 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 $H$ and $K$ are characteristic and maximal and $G$ 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.