Let $k$ be a local field with absolute Galois group $\Gamma_k$, inertia subgroup $I_k \subset \Gamma_k$, and residue field $\mathbb{F}$. Let $M$ be a finite Galois module over $k$.

The unramified Galois cohomology of $M$ in degree $i$ is defined to be $$H^i_{\mathrm{un}}(k,M) := \mathrm{Im}(H^i(\mathbb{F},M^{I_k}) \to H^i(k,M)).$$ My question is as follows. Let $$0 \to M_1 \to M_2 \to M_3 \to 0$$ be a short exact sequence of Galois modules. Then does this induce a long exact sequence $$0 \to H^0_{\mathrm{un}}(k,M_1) \to H^0_{\mathrm{un}}(k,M_2) \to H^0_{\mathrm{un}}(k,M_1)\to H^1_{\mathrm{un}}(k,M_1) \to \cdots$$ of unramified Galois cohomology groups?

If the modules $M_i$ are unramified, in the sense that $I_k$ acts trivially on each $M_i$, then this is clear. Otherwise the property is not so clear to me.

Generally I would appreicate any references on the basics of unramified cohomology, as it seems difficult to find the theory developed in a cohesive way in the literature.