I'm learning Galois/Group Cohomology and i've just seen that the second cohomology group $H^2(G,A)$ (constructed as the factor systems quotient splitting factors) classifies the group extensions of $G$ with abelian kernel $A$ up to equivalence.
My question is, given 2 extensions $$1\to A\to E\to G\to 1$$ and $$1\to A\to E'\to G\to 1$$ with cohomology classes $[c]$ and $[c']$ respectively ($[c]\ne [c']$), what is the extension (or the class of equivalent extensions) with cohomology class $[c]+[c']$ (assuming it is non-trivial)? Does it have a general form or it depends on the specific case?
I have not found it anywhere so I would appreciate some book where I can find it.
The extension corresponding to the sum of cohomology classes represented by the two original extensions is called the Baer sum of the two extensions.
It can be defined using just group theory without cohomology.