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.



Does a non-simple perfect group always have a maximal subgroup whose derived subgroup has nontrivial core?
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.


