Bar two-cocycles classify abelian-kernel extensions

For an extension E, apply AC to the nonempty fibers of $E\to G$ and set $s(1)=1$. Unique kernel coordinates define f by $s(g)s(h)=i(f(g,h))s(gh)$. Then $f(1,g)=f(g,1)=0$. Associativity of $s(g)s(h)s(l)$ gives $f(g,h)+f(gh,l)=gf(h,l)+f(g,hl)$, exactly $\delta f=0$. The action term follows from $s(g)i(a)s(g{)}^{-1}=i(ga)$, the prescribed action.

Conversely, for a normalized cocycle f define $E_f=A\times G$ with $(a,g)(b,h)=(a+gb+f(g,h),gh)$. The first coordinates of the two triple products differ by $f(g,h)+f(gh,l)-gf(h,l)-f(g,hl)=0$, so multiplication is associative. The identity is (0,1). The inverse is $(a,g{)}^{-1}=(-g^{-1}a-g^{-1}f(g,g^{-1}),g^{-1})$; the right product is the identity, and the left product is too since the cocycle identity gives $f(g^{-1},g)=g^{-1}f(g,g^{-1})$. Inclusion $a\mapsto (a,1)$ and projection to G are exact and conjugation induces ga.

A new normalized section $s_b(g)=i(b(g))s(g)$ changes f to $f_b=f+\delta b$, where $(\delta b)(g,h)=b(g)+gb(h)-b(gh)$. The map $E_f\to E_{f_b}$, $(a,g)\mapsto (a-b(g),g)$ is an isomorphism: substitution in the two multiplication laws gives first coordinate $a+gc+f(g,h)-b(gh)$ on both sides. It fixes A and G and has inverse adding b(g). If two normalized cocycles differ by a coboundary, the cochain b is normalized as well, since its coboundary at (1,g) equals b(1).

The map $E_f\to E$, $(a,g)\mapsto i(a)s(g)$, is a homomorphism by the factor-set equation; unique kernel coordinates in each fiber make it a bijection fixing A and G. Conversely any equivalence carries a chosen section to a section and preserves its factor set. Thus the two constructions induce inverse bijections on the quotient by coboundaries and on extension classes. By F2 this quotient is the indicated H2.

If the class is zero, choose a normalized b with $f+\delta b=0$; the section $s_b$ is then a homomorphism. A homomorphic section conversely has f=0 and hence zero class. For A=0 there is the unique extension G, and for G=1 the unique extension A; both have zero class. Only step 1.1 uses arbitrary choice; supplied sections suffice for an individual construction.