  In Part I, we use Steenrod's construction to prove that the quantum Coulomb
branch is a Frobenius-constant quantization. We will also demonstrate the
corresponding result for the $K$-theoretic version of the quantum Coulomb
branch. In Part II, we use the same method to construct a functor of
categorical $p$-center between the derived Satake categories with and without
loop-rotation, which extends the Frobenius twist functor for representations of
the dual group.
