Suppose I have a finite Galois extension of fields $K/k$, as well as a finite group $G$ with a surjection $f: G \rightarrow \mathrm{Gal}(K/k)$.

Finally, suppose I have an action $\sigma$ of $G$ on a vector space $V$ over $K$ by twisted endomorphisms, i.e. $\sigma(av) = f(\sigma)(a)\sigma(v)$.

How can I detect whether $G$ descends to a representation over $k$, i.e. when does there exist an untwisted representation of $G$ on a vector space $W$ such that $V = W \otimes_{k} K$ with the diagonal action?

I guess this is some group/Galois cohomology obstruction, but I wasn't sure how to set up the problem.