In Caruso's paper, "Operations in equivariant $Z/p$-cohomology," http://www.ams.org/mathscinet-getitem?mr=1684248, he shows that the integer-graded stable cohomology operations in $RO(\mathbb{Z}/p)$-graded cohomology (with constant $\mathbb{Z}/p$ Mackey functor coefficients) have image in the usual (non-equivariant) mod-$p$ Steenrod algebra consisting of only the identity and Bockstein operations.

I would like to know:

What do we know (if anything) about the rest of the cohomology operations, i.e. the ones which correspond to non-integer gradings and the unstable operations?

What do we know about the image of the $RO(\mathbb{Z}/p)$-graded (un)stable operations in the Steenrod algebra for Bredon cohomology?

Who is currently thinking about this stuff? (Besides me.)