We have the short exact sequence: $$1\rightarrow C\rightarrow \widetilde{G}\rightarrow G\rightarrow 1$$
Equipped witha map $i$ from $C$ to $\widetilde{G}$, and a map $p$ from $\widetilde{G}$ to G, such that $i(C)\subset Z(\widetilde{G})$
For each $g \in G$ fix a representative $\widetilde{g}\in p^{-1}(g)$, and show that the failure of the map; $$g\mapsto \widetilde{g}$$ to be a group homomorphism is measure by a map; $$\beta:G\times G\rightarrow C$$ That is, show that there is a unique $\beta$ with; $$\widetilde{g}\cdot\widetilde{h}=i(\beta(g,h))\widetilde{gh}$$ Finally, show that $\beta$ satisfies the 2-cocycle condition; $$\beta(g,h)\beta(gh,k)=\beta(g,hk)\beta(h,k)$$
First I want to show such a $\beta$ exists, then I'll assume there are 2 such $\beta$s, and show that they're equal. We have that $\beta(g,h)\in C$, and that $i(C)\in Z(\widetilde{G})\subset\widetilde{G}$
So, $i(\beta(g,h)\in \widetilde{G}$.
We also know that $\widetilde{g} \cdot \widetilde{h}=p^{-1}(g)p^{-1}(h)\in\widetilde{G}$, and that $\widetilde{gh}=p^{-1}(gh)\in\widetilde{G}$.
But from here, I have no idea where to go. I can't seem to figure out how to connect $\widetilde{g} \cdot \widetilde{h}$, and $\widetilde{gh}$. As for the second part, I have no idea. I'm assuming it reloies on the workings of the first part, so I included it just for completeness.
Thanks in advance for any help, I haven't seen central extensions before and am very new to the world of abstract algebra in general, so sorry if this is a trivial question.