  Let $G$ be a finite group and $E$ be an $H_\infty$-ring $G$-spectrum. For any
$G$-space $X$ and positive integer $m$, we give an explicit description of the
smallest Mackey ideal $\underline{J}$ in $\underline{E}^0(X\times B\Sigma_m)$
for which the reduced $m$th power operation $\underline{E}^0(X) \to
\underline{E}^0(X \times B\Sigma_m )/\underline{J}$ is a map of Green functors.
We obtain this result as a special case of a general theorem that we establish
in the context of $G\times\Sigma_m$-Green functors. This theorem also
specializes to characterize the appropriate ideal $\underline{J}$ when $E$ is
an ultra-commutative global ring spectrum. We give example computations for the
sphere spectrum, complex $K$-theory, and Morava $E$-theory.
