For a finite group G, we introduce the complete suboperad $Q_G$ of categorical G-Barratt-Eccles operad $P_G$. We prove that $P_G$ is not finitely generated, but generated and genuine $E_\infty$ G-operad (i.e., it $N_\infty$ includes all norms). G cyclic order 2 or 3, determine presentations object conclude with discussion algebras over $Q_G$, which call biased permutative equivariant categories.