Formula for second cohomology group for trivial group action in terms of Baer invariant and verbal factor group

General case
Suppose $$\mathcal{V}$$ is a subvariety of the variety of groups containing the variety of abelian groups. Suppose $$G$$ is a group (not necessarily in $$\mathcal{V}$$) and $$A$$ is an abelian group. Then, the second cohomology group for trivial group action $$H^2(G;A)$$ occurs in the following natural exact sequence:

$$0 \to H^2_{\mathcal{V}}(G/V(G);A) \to H^2(G;A) \to \operatorname{Hom}(\mathcal{V}M(G),A) \to ? \to H^3(G;A)$$

We are interested in the initial part of this, i.e.:

$$0 \to H^2_{\mathcal{V}}(G/V(G);A) \to H^2(G;A) \to \operatorname{Hom}(\mathcal{V}M(G),A)$$

The sequence splits (though the splitting is not necessarily natural), and we get:

$$H^2(G;A) \cong \operatorname{Hom}(\mathcal{V}M(G),A) \oplus \mbox{a subgroup of } H^2_{\mathcal{V}}(G/V(G);A) \mbox{ given by the image of the above map}$$

Here:


 * $$H^2_{\mathcal{V}}(G/V(G);A)$$ is defined as follows: $$H^2_{\mathcal{V}}$$ denotes the second cohomology group restricted to a subvariety, $$V(G)$$ is the verbal subgroup of $$G$$ with respect to $$\mathcal{V}$$, and we take the action of $$G/V(G)$$ on $$A$$ to be trivial. The image of $$H^2_{\mathcal{V}}(G/V(G);A)$$ in $$H^2(G;A)$$ is the subgroup $$H^2_{\mathcal{V}}(G;A)$$.
 * $$\operatorname{Hom}(\mathcal{V}M(G),A)$$ is the group of abelian group homomorphisms (under pointwise addition) from the Baer invariant $$\mathcal{V}M(G)$$ to $$A$$.
 * The image of $$H^2(G;A)$$ in $$\operatorname{Hom}(\mathcal{V}M(G),A)$$ is the second cohomology group up to isologism for $$G$$ acting trivially on $$A$$. In other words, the fibers of this mapping are precisely the equivalence classes of extensions under isologism.

Note that for some choices of subvariety $$\mathcal{V}$$, such as if $$\mathcal{V}$$ is the variety of abelian groups, it has been shown that the truncated sequence is exact, i.e., the map from $$H^2(G;A)$$ to $$\operatorname{Hom}(\mathcal{V}M(G),A)$$ is surjective, or equivalently, the second cohomology group up to isologism is the entire group of homomorphisms $$\operatorname{Hom}(\mathcal{V}M(G),A)$$. However, this is not true for all subvarieties of the variety of groups.

Related facts

 * Formula for second cohomology group for trivial group action in terms of second homology group and abelianization is the particular case where we use the variety of abelian groups.
 * Formula for second cohomology group for trivial group action in terms of nilpotent multiplier and lower central series quotient