Second cohomology group for trivial group action is internal direct sum of symmetric and cyclicity-preserving 2-cocycle subgroups if acting group is elementary abelian 2-group and every element of order two in the base group is a square

In terms of internal direct sums
Suppose $$G$$ is an $$2$$-group that is an fact about::elementary abelian group and $$A$$ is an abelian group with the property that every element of order $$2$$ in $$A$$ is a square. Then, we have:

$$\! H^2(G,A) = H^2_{sym}(G,A) + H^2_{CP}(G,A)$$

In words, the fact about::second cohomology group for trivial group action of $$G$$ on $$A$$ is the internal direct product (internal direct sum) of the subgroup generated by the images of symmetric 2-cocycles (which corresponds to the extensions that are abelian) and the fact about::cyclicity-preserving subgroup of second cohomology group for trivial group action (which is the image in cohomology of the group of cyclicity-preserving 2-cocycles).

In terms of existence of generalized Baer cyclicity-preserving Lie ring
Suppose $$G$$ is an $$2$$-group that is an elementary abelian group and $$A$$ is an abelian group with the property that every element of order $$2$$ in $$A$$ is a square. Then, every central extension with base $$A$$ and quotient $$G$$ has its own generalized Baer lie ring via the fact about::cocycle skew reversal generalization of Baer correspondence.