(Not sure why this question comes up naturally. The more interesting question, and the one analogue to the kernel of restriction, is to ask what is the cokernel of corestriction. That turns up a lot. Like in class field theory where the cokernel of the norm map is a central question, while the kernel of the norm map is large and not of the same level of interest.)
The Tate spectral squence (Cohomology of number field, printed version, Theorem 2.1.11) involves the corestriction map (loc cit Theorem 2.1.12). The resulting five term exact sequence answers the question. Let $K/k$ be a finite extension of local or global fields with Galois group $G$ and let $M$ be a Galois module for $\operatorname{Gal}(\bar k/k)$ whose order is a unit in $k$. Then this is the dual of the said 5-term exact sequence: $$ 0 \leftarrow H_1\bigl(G, H^2(K,M)\bigr) \leftarrow H^1(k,M)\overset{\operatorname{cor}}{\leftarrow} H_0\bigl(G,H^1(K,M)\bigr) \leftarrow H_2\bigl(G,H^2(K,M)\bigr) \leftarrow H^2(k,M) $$ In the case of a global field, I would switch to the restricted cohomology with respect to a finite set $S$ of places such that $K/k$ and $M$ are unramified outside $S$ and $S$ contains all places above primes divising the order of $M$. For infinite $G$, one has to replace the inner cohomology groups by limits taken over corestriction maps for finite subextensions as in loc.cit.