Apart from the direct products, what are some "interesting" or "naturally occurring" examples of extensions
$$
1 \to N \to G \to Q \to 1
$$
of finite groups such that *neither* $N$ *nor* $Q$ is solvable?


I feel a bit stupid for asking the question, but I don't think I know a single example that isn't split.


Just to set the bar pretty low: essentially the only example I can think of is to take the wreath product $\mathfrak{S}\_n\wr\mathfrak{S}\_m$ (group of permutations of $mn$ elements preserving a partition into $m$ sets of $n$), for $m,n\geq 5$, which is a semidirect product $(\mathfrak{S}\_n)^m \rtimes \mathfrak{S}\_m$. Anything which is not a trivial variation on this construction interests me.



Examples of extensions of non-solvable groups by one another
Here is a way you can construct nonsplit examples in which all composition factors are nonabelian.


Many finite nonabelian simple groups do not split over their automorphism groups. The smallest such example is the group often known as $M\_{10}$, which is the point stabilizer in the Mathieu group $M\_{11}$. It has order $720$ and is the extension of ${\rm PSL}(2,9)$ by a product of a field and a diagonal automorphism. So it has a normal subgroup $H$ of index $2$ with $H \cong A\_6 \cong {\rm PSL}(2,9)$.


Now $M\_{10}/H$ is of course solvable so we don't have an example yet, but we can use a wreath product type construction to get what we want.


Let $K$ be any finite nonabelian simple group, and let $S$ be any subgroup of order $2$ in $K$. Let $P$ denote the image of the permutation action of $K$ on the cosets of $S$. So, for example, if $K=A\_5$ then $P$ is a subgroup of $S\_{30}$.


Now the permutation wreath product $W$ of $H$ by $P$ is of course a split extension. $1 \to H^{|K:S|} \to W \to P \to 1$.


But there are in general other extensions of this type, and there is a theorem that says that the equivalence classes of such extensions are in one-one correspondence with the extensions of $H$ by $S$. So, if there is a nonsplit extension of $H$ by $S$, then there is also one of $H^{|K:S|}$ by $P$. (I will look for a reference for that result later.)


For example, if we take $H={\rm PSL}(2,9) \cong A\_6$ as above, and $K=A\_5$, then we get a nonsplit extension with normal subgroup $A\_6^{30}$ and quotient $A\_5$.


I found a reference for the result about wreath product type extensions in an old paper of my own. It is proved as Theorem 1 of D.F. Holt, Embeddings of group extensions into Wreath products,
 Quar. J. Math. (Oxford) 29 (1978), 463--468. But I suspect it was known earlier. The main result of that paper is a generalization of the Krasner Kaloujnine theorem about embedding group extensions into standard wreath products.


