Artin-Schreier Root Stacks and lifts of group actions

Let $G$ be a connected affine algebraic group defined over a field of positive characteristic. We prove that the action of $G$ on a smooth projective variety can be lifted to its associated Artin-Schreier root stacks, whenever $G$ has no non-trivial characters. The existence of a $G$-linearization on a certain tautological invertible sheaf on such Artin-Schreier root stacks is also shown.