On nonsplit noncentral extension of finite simple groups

Question

Let $G$ be a finite group and $N$ be its minimal normal subgroup such that $G/N$ is a finite simple group.

It is well know that if $G=G'$ and $N=Z(G)$, then $N=\Phi(G)$.

My Question is: Is it true that if $G=G'$ and $N=\Phi(G)$, then $N=Z(G)$?

Any examples and explanations are appreciated.

Answer 1

No, any nonsplit extension of a nontrivial irreducible module by a simple group is a counterexample.

For example, we could take $G$ to be a nonsplit extension ${2^3} {^\cdot} L_3(2)$ of the natural module $N$ for $S = {\rm GL}(3,2)$ by $S$. So $|N|=8$, $|S| = |G/N| = 168$ and $|G| = 1344$.