We study the indexing systems that correspond to equivariant linear
isometries operads and infinite little discs operads. When $G$ is a finite
abelian group, we prove that a $G$-indexing system is realized by a little
discs operad if and only if it is generated by cyclic $G$-orbits. When $G =
C_n$ is a finite cyclic group, and $n$ is either a prime power or $n = pq$ for
primes $3 < p < q$, we prove that a $G$-indexing system is realized by a linear
isometries operad if and only if it satisfies Blumberg and Hill's horn-filling
condition.
We also develop equivariant algebra, at times necessary for, and at times
inspired by the work above. We introduce transfer systems, a finite
reformulation of the data in an indexing system, and we construct image and
inverse image adjunctions for transfer systems that are analogous to
equivariant induction, restriction, and coinduction. We construct derived
induction, restriction, and coinduction functors for $N_\infty$ operads, and we
prove that they correspond to their algebraic counterparts for injective maps.