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
In words, the Second cohomology group for trivial group action (?) of on 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 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 is an -group that is an elementary abelian group and is an abelian group with the property that every element of order in is a square. Then, every central extension with base and quotient has its own generalized Baer lie ring via the Cocycle skew reversal generalization of Baer correspondence (?).