Second cohomology group for trivial Lie ring action of V4 on Z2

Description of the group
We consider here the second cohomology group for trivial Lie ring action of the specific information about::Klein four-group on specific information about::cyclic group:Z2, i.e.,

$$\! H^2(G;A)$$

where $$G \cong V_4 \cong \mathbb{Z}_2 \times \mathbb{Z}_2$$ and $$A \cong \mathbb{Z}_2$$.

The cohomology group is isomorphic to elementary abelian group:E8.

Summary
Each element of the second cohomology group corresponds to a Lie ring extension with base ideal an abelian Lie ring isomorphic to cyclic group:Z2 in the center and the quotient Lie ring an abelian Lie ring whose additive group is isomorphic to Klein four-group. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each extension Lie ring is $$2 \times 4 = 8$$.

Further, the minimum size of generating set of the extension group is at least equal to 2 (the minimum size of generating set of the quotient Klein four-group) and at most equal to 3 (the sum of the minimum size of generating set for the normal subgroup and quotient group).