 First Galois cohomology unramified outside a finite set

For a finite Galois module $M$ unramified outside a finite set $S$, define $H_S^1(K,M)$ as the classes in Galois cohomology whose restriction to inertia is zero at every finite prime outside $S$. This means zero as a cohomology class, rather than zero for every possible representative. Passing to a finite Galois extension on which $M$ is constant reduces the finiteness problem to characters with bounded ramification and a finite restriction kernel.