Let $G$ be a locally compact group, let $N$ be a closed normal subgroup of $G$, and let $\sigma\colon G/N\to G$ be a cross section. Let us define $\alpha\colon G/N\times G/N \to N$ by the formula $$ \sigma(x)\sigma(y)\alpha(x,y)=\sigma(xy), $$ for all $x,y\in G/N$.
Then $\alpha$ records how far $\sigma$ is from being multiplicative. Is there a standard name for $\alpha$? It seems to be a variant of a $2$-cocycle, but it is not the usual $2$-cocylce associated to a cross section.
I believe David Roberts is correct that $\alpha$ is a $2$-cocycle representing an element of $H^2(G/N,N)$, and is the usual $2$-cocycle associated to this extension.
The confusion comes from the fact that $\omega$ is not a $2$-cocycle. Rather, it is called a cocycle because it can be thought of as a $1$-cocycle representing a class in $H^1 \bigl( G, \mathcal{F}(G/N; N) \bigr)$, where $\mathcal{F}(G/N; N)$ is the space of functions from $G/N$ to $N$. This is a group under pointwise multiplication, and $G$ acts on it via $\varphi^g(x) = \varphi(gx)$.
More precisely, define $\omega_g(x) = \omega(g,x)$, so $\omega_g \in \mathcal{F}(G/N; N)$ for each $g \in G$. Then $$ \sigma(ghx) \, \omega(gh,x) = gh \, \sigma(x) = g \cdot \sigma(hx) \, \omega(h,x) = \sigma(ghx) \, \omega(g, hx) \, \omega(h,x) ,$$ so $\omega_{gh} = (\omega_g)^h \cdot \omega_h$. This is precisely what it means to say that $\omega_g$ is a $1$-cocycle.
That $\omega$ is a $1$-cocycle (in the more general context of $G$-actions on principal bundles) is mentioned at the bottom of page 66 of R.J.Zimmer's book Ergodic Theory and Semisimple Groups.
