Assume Q and N are groups, with N abelian and $w:Q\rightarrow \text{Aut}(N)$ is a fixed homomorphism. Moreover, we assume $f \in Z^2(Q,N,w)$ is a normalized 2-cocycle and G is the group whose underlying set is the Cartesian product $Q \times N$ and whose product is given by $$(q_1,n_1).(q_2.n_2)=(q_1q_2,f(q_1,q_2)+w_{q_2}(n_1)+n_2)$$ therefore G is an extension of N by Q where $l:N\rightarrow G$ and $\phi:G\rightarrow Q$ the natural maps defined by $l(n)=(1_Q,n)$ and $\phi(q,n)=q$. show that following statements are equivalent.
(1) there exists a homomorphic section $\theta:Q\rightarrow G$.
(2) The image $l(N)=\tilde{N}$ has a complement in G.
(3) The image $[f]$ of $f$ in the second cohomology group $H^2(Q,N,w)$ is zero.
I showed that (1) is equivalent to (2).I showed that if $\theta$ be homomorphic then $\theta(Q)$ is complement of $\tilde{N}$. Conversely, if $\tilde{N}$ has complement $C$, then I showed that there is an isomorphic from $Q$ to $C$ which is the $\theta.$
For part (3) I have no idea.