I'm trying to understand unramified Galois cohomology of number fields a bit better.
Set-up: Let $k$ be a number field and $S$ a finite set of places of $k$ which contains all the archimedean places. Fix an algebraic closure $\bar k$ of $k$ and denote by $k_S$ the maximal field extension of $k$ which is unramified outside of $S$. Let $G$ be an algebraic group over $k$, not necessarily finite nor abelian (this is important for the application which I have in mind.)
I shall say that a class $c \in H^1(k,G(\bar k))$ is unramified at a place $v$ of $k$ if it lies in $$\ker\left(H^1(k,G(\bar k)) \to H^1(I_v,G(\bar k)\right),$$ where $I_v$ denotes the inertia group at $v$. I shall denote by $H^1_S(k,G(\bar k))$ the collection of elements of $H^1(k,G(\bar k))$ which are unramified outside of all places of $S$.
This looks like it should be closely related to $H^1(k_S/k, G(k_S))$. My question is as follows
How are $H^1_S(k,G(\bar k))$ and $H^1(k_S/k, G(k_S))$ related? For example, are they equal? Is one naturally contained in the other?
I'm willing to enlarge the set of places $S$ if necessary to include any "bad" primes.