Suppose that $G$ is a profinite group acting faithfully on a field $L$ and assume moreover that the action is admissible, that is, that every $l \in L$ is stabilized by an open subgroup of $G$. If that's the case, then we have Galois descent, that is, there's an equivalence
$Vect_{L}^{G} \simeq Vect_{L^{G}}$
between the category of $L$-vector spaces equipped with a compatible admissible action of $G$, and the category of vector spaces over the fixed field $L^{G}$. In particular, the category $Vect_{L}^{G}$ is semisimple.
Question: What happens if we remove the admissibility assumption? To give a precise example, suppose that $G$ acts faithfully and continuously on a complete discrete valuation ring $R$ with a finite residue field $R / \mathfrak{m} \simeq k$, where we consider the latter as equipped with the profinite topology coming from the isomorphism $R \simeq \varprojlim R / \mathfrak{m}^{n}$. If that's the case, then the action of $G$ extends to a continuous action on the fraction field $K = (R)$.
Do we have any form of Galois descent in the case of such an action? How is the category of (say, finite-dimensional) $K$-vector spaces equipped with a compatible continuous $G$-action related to the category of vector spaces over the fixed field $K^{G}$? Is the category of such representations necessarily semisimple?
