5 changed title, improved formatting

Let F $F$ be a local field of characteristic zero(for zero (for simplicity), E $\overline{F}$ an algebraic closure of F $F$ and L/F $L/F$ a fixed finite Galois extension. If G $G$ is a linear algebraic group defined over F, $F$, then the Galois cohomology group $H^1(F,G)$ can be defined as a direct limit of $H^1(K/F,G)$, where K $K$ runs through finite Galois subextensions of E.$\overline{F}$.
Now the question is: under what conditions , is this direct limit is just $H^1(L/F,G)$?
I guess there might be restrictions on both $L$ and $G$.