Consider an extension of groups $$(E)\ :\ 1\to N\to G\stackrel{\pi}{\longrightarrow} Q\to 1$$ and assume 


1. $N$ is abelian,
2. $Q$ is finite,
3. for any Sylow subgroup $S\_p$ of $Q$, the pullback $(E\_p)\ :\ 1\to N\to \pi^{-1}(S\_p)\to S\_p\to 1$ is split.


Then $(E)$ is split. 


[A proof can be written using only elementary homological algebra plus Cartan-Eilenberg double coset formula.]


If we drop the first hypothesis, the conclusion doesn't hold in general. The simplest example seems to be the extension
$$1\to F\_2\to PSL\_2(\mathbf Z)\simeq \mathbf Z/2\ast\mathbf Z/3\to PSL\_2(\mathbf F\_2)\simeq\mathfrak S\_3\to 1$$


Even if we insist that $N$ be finite solvable, there are counter-examples (e.g. take the above with $PSL\_2(\mathbf Z/8)$ instead of $PSL\_2(\mathbf Z)$). 



> 
> Assume $N$ is solvable and $\pi$ is split over all normalizers of Sylow subgroups in $Q$. Is $\pi$ necessarily split ?
> 
> 
> 



Extension splitting over Sylow subgroups
This didn't seem likely, so I looked for a counterexample and found one after a short search. Clearly $Q$ must not have any normal Sylow subgroups, so the first example to try is $Q=S\_4$. Then $N$ must not be abelian, so I tried $N=Q\_8$ and found an example.


The example is $\mathtt{SmallGroup}(192,988)$ in GAP or Magma. This has seven generators $x\_1,\ldots,x\_7$, and the normal subgroup isomorphic to $Q\_8$ is $\langle x\_2, x\_3x\_7 \rangle$.


Checking that the extension is nonsplit but it splits over the normalizers of Sylow $2$- and $3$-subgroups is routine in GAP or Magma, but I can tell you the commands if it would help.


Magma code follows. Note that the Sylow $2$-subgroup is self-normalizing in $G$. The normalizer of the Sylow $3$-subgroup of $G$ contain $N$, so it is equal to the inverse image of the normalizer in $Q = G/N$ of a Sylow $3$-subgroup.



```
> G := SmallGroup(192,988);
> N := sub< G | G.2, G.3*G.7 >;
> #N;
8
> IsNormal(G,N);
true    
> #Complements(G,N);
0
> P := Sylow(G,2);
> P eq Normalizer(G,P);
true
> #Complements(P,N);
4
> NQ := Normalizer(G, Sylow(G,3));
> N subset NQ;
true
> #Complements(NQ,N);
1

```

