Is there a proper subgroup of any symmetric group that contains both odd and even permutations and is non-solvable?

Question

Does there exist an $n$ such that there is a subgroup $G \subset S_n$ where

1. G is non-solvable, and
2. G contains both an odd and an even permutation?

Answer 1

Taking $\,S_n\,$ as a group of permutations on the set $\,\{1,2,...,n\}\subset\Bbb N\,$ , take

$$G:=\{\sigma\in S_n\;:\;\sigma(n)=n\}\cong S_{n-1}\,\,,\,\,n\geq 6$$

Answer 2

Take any nonsolvable group $G$ of order $n=2^am$, where $a>0$ and $m>1$ is odd. By Cayley's theorem, $$G\leqslant S_n < S_{n+1}.$$ $G$ contains both even and odd permutations by Cauchy's theorem and is a proper nonsolvable subgroup of $S_{n+1}$.