Kummer theory treats Galois extensions of exponents that are not divisible by the characteristic. Artin-Schreier and Witt extend this theory for Galois extensions of exponents $p^r$ in characteristic $p$ ($p$ prime, $r \in \mathbb{N}$).
Is there any theory that extends/combines this theory to Galois extensions of exponent $p^r m$, where $m$ is not divided by $p$?
From chapter 10 of Bosch's book "Algebra" I know that general Kummer theory can be applied to Galois groups $G$ if there exist a continous $G$-module $A$ and a surjective $G$-homomorphism $\phi: A \rightarrow A$ with certain properties. For Galois extensions of exponent $p^r m$ Witt rings are used to define $A$. Is there something similar for extensions of exponent $p^r m$?