Please accept my apologies for another elementary group theory question.

Let $$N\hookrightarrow E \twoheadrightarrow G$$ be a group extension such that the induced outer action $\psi\colon\thinspace G\to \mathrm{Out}(N)$ is trivial. Is it necessarily true that $N$ is central in $E$?

I have read comments on this site to this effect, but have been unable to come up with a proof. All I am seeing is that the conjugation action $\psi\colon\thinspace E\to \mathrm{Aut}(N)$ has image in the group $\mathrm{Inn}(N)$ of inner automorphisms.

On the other hand, work of Eilenberg and Mac Lane (as summarized in Brown's book "Cohomology of Groups", Section IV.6) shows that extensions as above are classified by $H^2(G;C)$, where $C$ denotes the centre of $N$ regarded as a trivial $G$-module, and therefore by central extensions of the form $$C\hookrightarrow A \twoheadrightarrow G.$$ This seems to suggest that the above statement is indeed true.