See 15.21 in pages 177-190 of here, where I collected the results on extensions of groups and Lie groups that I could find. 15.24 summarizes your situation quite clearly: We have $\text{Inn}(N)= N/Z(N)$ where $Z(N)$ is the center of $N$. Then you have a central extension $$Z(N)\to E \to G\times\text{Inn}(N)$$ described by the cohomology class in $H^2(G,Z(N))$ which projects to your extension $$N\to E\to G$$ with identity on $E$.
If somebody manages to insert the diagram, I would be grateful.