  $N_\infty$-operads are an equivariant generalization of $E_\infty$-operads
introduced by Blumberg and Hill to study structural problems in equivariant
stable homotopy theory. In the original paper introducing these objects,
Blumberg and Hill raised the question of classifying $N_\infty$-operads that
are weakly equivalent to a particularly nice kind of $N_\infty$-operad called a
linear isometries operad. For some groups there is a known classification of
linear isometries operads up to weak equivalence in terms of certain
combinatorially defined objects called saturated transfer systems, but this
classification is known to be invalid in general. Various authors have made
incremental progress on understanding the domain of validity for this
classification, but even among cyclic groups the validity is unknown in
general. We determine essentially all the finite Abelian groups for which the
classification is valid using techniques from algebra and extremal
combinatorics.
