Edit: Diagram added, more details added.

See 15.21 in pages 177-190 of [here][1], 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 $Z(N)$.


  [1]: http://www.mat.univie.ac.at/~michor/dgbook.pdf