Second cohomology group up to isologism

Definition
Suppose $$G$$ is a group and $$A$$ is an abelian group. Specify a variety $$\mathcal{V}$$ of groups that contains the variety of abelian groups. The second cohomology group up to isologism is the quotient of the usual second cohomology group by the following equivalence relations: two extensions $$E_1, E_2$$ are isologic as extensions if there is an defining ingredient::isologism between them that is compatible with the identity maps for $$G$$ and $$A$$.

Relation with formula for second cohomology group
Consider the case that the action is trivial, i.e., we are looking at the second cohomology group for trivial group action.

Then, the second cohomology group up to isologism can be identified with a subgroup of the group:

$$\operatorname{Hom}(\mathcal{V}M(G),A)$$

where $$\mathcal{V}M(G)$$ is the Baer invariant with respect to $$\mathcal{V}$$.

In the case that $$\mathcal{V}$$ is precisely the variety of abelian groups, isologism becomes isoclinism. It turns out in this case that it is the full homomorphism group.

It also occurs as one of the terms in the formula for second cohomology group for trivial group action in terms of Baer invariant and verbal factor group.