  We study actions of pointed Hopf algebras in the $\ZZ$-graded setting. Our
main result classifies inner-faithful actions of generalized Taft algebras on
quantum generalized Weyl algebras which respect the $\ZZ$-grading. We also show
that generically the invariant rings of Taft actions on quantum generalized
Weyl algebras are commutative Kleinian singularities.
