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We study the indexing systems that correspond to equivariant Steiner andlinear isometries operads. When $G$ is a finite abelian group, we prove that a$G$-indexing system is realized by a Steiner operad if and only if it isgenerated by cyclic $G$-orbits. When $G$ is a finite cyclic group, whose orderis either a prime power or a product of two distinct primes greater than $3$,we prove that a $G$-indexing system is realized by a linear isometries operadif and only if it satisfies Blumberg and Hill's horn-filling condition.  We also repackage the data in an indexing system as a certain kind of partialorder. We call these posets transfer systems, and we develop basic tools forcomputing with them.

Characterizations of equivariant Steiner and linear isometries operads