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.

Edit: Diagram added, more details added.

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 mapping of extensions $$ \begin{array}{ccccc} Z(N) & \xrightarrow{i|_{Z(N)}} & E & \xrightarrow{\theta} & G\times\text{Inn}(N) \newline \downarrow & & \downarrow & & \downarrow \newline N & \xrightarrow{i} & E & \xrightarrow{p} & G \end{array} $$ where the down arrows are inclusion, identity, and first projection, and where $\theta(x)=(p(x),\text{Conj}_x|_N)$. The first line is a central extension since $G\times \text{Inn}(N)$ acts trivially on $Z(N)$.