Group cohomology question, trivial Galois action on discrete Galois module means we can say what about kernel of map

If the $G_K$-action on $M$ is trivial, then $$H^1(K,M)=\mathrm{Hom}(G_K,M),$$ and by Chebotarev's density theorem $$ F^1(K,M)=0.$$

For details see Lemma 1.1(i) of Sansuc's paper.

For a refined statement, see also https://www.mathi.uni-heidelberg.de/~schmidt/NSW2e/index-de.html (9.1.9) (i).