Sorry about that, I formulated another question

Hi Pete, very cool result!

Thanks Jack, I understood your point, there was a logical mistake in the way I set things up. I just edited it

So what I meant in my question was that if $H$ is a primitive group which acts on $\\{1,2,\ldots,N\\}$ and $H$ contains a $\ell$-cycle with $\ell\leq N-7$ then $H$ contains $A_N$. This comes from the fact that there are no $6$-ply transitive permutation groups outside $A_N$ and $S_N$.

change $\rtimes$ by $\ltimes$

Well take $H=S_n\rtimes\mathbf{F}_2^n$ with $N=2n$, this certainly have an onto group homomorphism to $S_n$.