We advance the foundational study of be Nardin-Shah's $\infty$-category of $G$-operads and their associated $\infty$-categories of algebras. In particular, we construct the underlying $G$-symmetric sequence of a (one color) $G$-operad, yielding a monadic functor; we use this to lift Bonventre's genuine operadic nerve to a conservative functor of $\infty$-categories, restricting to an equivalence between categories of discrete $G$-operads. Using this, we extend Blumberg-Hill's program concerning $\mathcal{ N}_\infty$-operads to arbitrary sub-operads of the terminal $G$-operad, which we show are equivalent to weak indexing systems. We then go on to define and characterize a homotopy-commutative and closed Boardman-Vogt tensor product on $\mathrm{ Op}_G$; in particular, this specializes to a $G$-symmetric monoidal $\infty$-category of $\mathcal{ O}$-algebras in a $G$-symmetric monoidal $\infty$-category whose $\mathcal{ P}$-algebras are objects with interchanging $\mathcal{ O}$-algebra and $\mathcal{ P}$-algebra structures.
Categories: math.CT math.AT