Total power operations in spectral sequences

We describe how power operations descend through homotopy limit spectral sequences. We apply this to describe how norms appear in the $C_2$-equivariant Adams spectral sequence, to compute norms on $\pi_0$ of the equivariant $KU$-local sphere, and to compute power operations for the $K(1)$-local sphere. An appendix contains material on equivariant Bousfield localizations which may be of independent interest.