  Continuing our previous work on graded twisting of Hopf algebras and monoidal
categories, we introduce a graded twisting construction for equivariant
comodule algebras and module categories. As an example we study actions of
quantum subgroups of $G\subset SL_{-1}(2)$ on $K_{-1}[x,y]$ and show that in
most cases the corresponding invariant rings $K_{-1}[x,y]^G$ are invariant
rings $K[x,y]^{G'}$ for the action of a classical subgroup $G'\subset SL(2)$.
As another example we study Poisson boundaries of graded twisted categories and
show that under the assumption of weak amenability they are graded twistings of
the Poisson boundaries.
