  In this paper, we prove a conjecture of Blumberg and Hill regarding the
existence of $N_\infty$-operads associated to given sequences $\mathcal{F} =
(\mathcal{F}_n)_{n \in \mathbb{N}}$ of families of subgroups of $G\times
\Sigma_n$. For every such sequence, we construct a model structure on the
category of $G$-operads, and we use these model structures to define
$E_\infty^{\mathcal{F}}$-operads, generalizing the notion of an
$N_\infty$-operad, and to prove the Blumberg-Hill conjecture. We then explore
questions of admissibility, rectification, and preservation under left
Bousfield localization for these $E_\infty^{\mathcal{F}}$-operads, obtaining
some new results as well for $N_\infty$-operads.
