In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact. 


$\textbf{Claim}$: If $N$ is a normal subgroup, minimal among normal subgroups, of a group $G$ not contained in the Frattini's subgroup of $G$, then $G$ is a semi-direct product over $N$, i.e. there exists $H$ subgroup of $G$ with $H \cap N=\{id\}$ and $HN=G$. 


Now, if $N$ is a simple non-abelian normal subgroup of $G$, the above Claim implies that $G$ is a semi-direct product over $N$. Indeed the Frattini of $G$ is nilpotent, and since $N$ is non-abelian, it cannot be contained in a nilpotent group, otherwise $N$ itself would be nilpotent, and nilpotent and simple implies having prime order and hence abelian. Moreover $N$ is clearly minimal, since $N$ itself does not have non-trivial proper normal subgroup. But then I see a clear contradiction with Lemma 4.3 of this paper of Lucchini, Menegazzo and Morigi <https://projecteuclid.org/download/pdf_1/euclid.ijm/1258488162>.


Could somebody explain (in case I did not perform a mistake in reasoning) which of the two publications has a mistake? 



About normal minimal subgroups not in the Frattini
I think it is the claim from exercise 2 of Neukirch et al which is incorrect. I presume that all groups are meant to be finite in this question. The claim from exercise 2 is OK when $N$ is solvable (for then $N$ is an elementary Abelian $p$-group for some prime $p$ and there is a maximal subgroup $H$ of $G$ which does not contain $N$. Then $H \cap N$ is normal in $N$ ( as $N$ is Abelian), and also normal in $H$, so $H \cap N$ is normal in $\langle H,N \rangle= G.$ By minimality of $N$ among (non-trivial) normal subgroups, we have either $H \cap N = N$ or $H \cap N =1.$ But we can't have $H \cap N = N$ as $H$ does not contain $N$).
 On the other hand, there are finite non-Abelian simple groups which are not complemented in their automorphism groups, and the example of Lucchini et al is one of these. Any such example contradicts the claim of Neukirch et al ( if indeed they do not have the restriction that $N$ is assumed solvable).


