I think the point is that, forgetting the final carry, the group of $n$-digit binary words is isomorphic to $C_{2^n}$. In the simplest case, the group of 2-digit binary words is isomorphic to $C_4$, which is built as a nontrivial extension $$ 0 \to C_2 \to C_4 \to C_2 \to 0 $$ The 2-cocycle you mention is the one corresponding to this extension. In general, $C_{2^n}$ is built up as an iterated extension of $C_2$'s in the same way, with each carry being the associated 2-cocycle. If we want to avoid forgetting the final carry, we can take the limit of the whole system to get the 2-adics $\mathbb{Z}_2$. The integers $\mathbb{Z}$ sit inside this as the subgroup of "finite words" (words whose digits are eventually 0 as we read right to left)

I think the point is that, forgetting the final carry, the group of $n$-digit binary words is isomorphic to $C_{2^n}$. In the simplest case, the group of 2-digit binary words is isomorphic to $C_4$, which is built as a nontrivial extension $$ 0 \to C_2 \to C_4 \to C_2 \to 0 $$ The 2-cocycle you mention is the one corresponding to this extension. In general, $C_{2^n}$ is built up as an iterated extension of $C_2$'s in the same way, with each carry being the associated 2-cocycle. If we want to avoid forgetting the final carry, we can take the limit of the whole system to get the 2-adics $\mathbb{Z}_2$. The natural numbers $\mathbb{N}$ sit inside this as the submonoid of "finite words" (words whose digits are eventually 0 as we read right to left)