Series-equivalent not implies automorphic

Statement
It is possible to have a group $$G$$ and normal subgroups $$H$$ and $$K$$ of $$G$$ that are fact about::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 $$K$$.

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

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