Let $G \neq 1$ be a finite perfect group which is not simple.
Is it true that $G$ necessarily has a maximal subgroup whose derived subgroup
has nontrivial core in $G$?
Remark 1: This holds for all such $G$ of order less than 100000.
Remark 2: In case the answer is negative, I would mainly be interested
in a counterexample with nontrivial Frattini subgroup.
Best Answer
The answer to the title question is 'No.' Following YCor's comment, an example is furnished by $S=J_1$, the smallest Janko sporadic group, and its complex irreducible character $\chi$ of degree 76 (in Atlas notation, 76a). I have checked by hand (hopefully correctly), using the Atlas, that $(\chi\downarrow M,1_M)>0$ for all maximal subgroups $M$ of $J_1$. For any prime $p$ not dividing $|J_1|=2^3.3.5.7.11.19$, let $N_p$ be an $F_pS$-module affording the mod-$p$ reduction of $\chi$. Then, as YCor points out, the semidirect product $N_pS$ has the property that for every maximal subgroup $H\le N_pS$, $[H,H]$ does not contain $N_p$.
However, this example has trivial Frattini subgroup.