  For any $E_\infty$ ring spectrum $E$, we show that there is an algebra
$\mathrm{Pow}(E)$ of stable power operations that acts naturally on the
underlying spectrum of any $E$-algebra. Further, we show that there are maps of
rings $E \to \mathrm{Pow}(E) \to \mathrm{End}(E)$, where the latter determines
a restriction from power operations to stable operations in the cohomology of
spaces. In the case where $E$ is the mod-$p$ Eilenberg-Mac Lane spectrum, this
realizes a natural quotient from Mandell's algebra of generalized Steenrod
operations to the mod-$p$ Steenrod algebra. More generally, this arises as part
of a classification of endomorphisms of representable functors from an
$\infty$-category $\mathcal{C}$ to spectra, with particular attention to the
case where $\mathcal{C}$ is an $\mathcal{O}$-monoidal $\infty$-category.
