Regularity of quotients by an automorphism of order $p$

Let $B$ be a regular local ring and $G\subset \text{Aut}(B)$ a finite group of local automorphisms. Assume that $G$ is cyclic of prime order $p$, where $p$ is equal to the residue characteristic of $B$. We give conditions under which the ring of invariants $A=B^G$ is again regular.