This question is not about odd and even permutations. Rather, I would like to understand the subgroup of the symmetry group $S_n$ that preserves the parity of all numbers $1, 2, \ldots, n$. So, if e.g. $m$ is odd, and $\sigma$ is a permutation in this subgroup, then $\sigma(m)$ is also odd.
As an example, the only members of $S_3$ that have this property are $()$ and $(13)$. I believe for $S_4$ this group has members $()$, $(24)$, $(13)$, and $(13)(24)$. Seems like this should be something that has been studied in the past. If there are any good resources on this topic, or if you know of a name for this permutation group which I could search up, please do let me know.
Let $M$ be the set of even numbers the permutations act on and $N$ be the set of odd numbers the permutations act on. Then the subgroup in question is
$$S_M\times S_N.$$