I think the answers for the first few degrees ($n$) are:

$n=2$, $S_2$

$n=3$, $S_3,A_3$

$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group)

$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ($Fr_5$ is a Frobenius group)

What results do we have for higher orders and are there any results for a general $n$?

I think the answers for the first few degrees ($n$) are:

$n=2$, $S_2$

$n=3$, $S_3,A_3$

$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group)

$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ($Fr_5$ is a Frobenius group)

What results do we have for higher orders and are there any results for a general $n$?

Edit: Sorry, I have aptly demonstrated that I am still rough on some of my concepts here. I believe I was thinking about the case where $f(x)\in \mathbb{Z}[x]$ and the base field is a finite extension of the rational numbers. I hope I have gotten it right this time and the above answers are right for this situation.

I think the answers for the first few degrees ($n$) are:

$n=2$, $S_2$

$n=3$, $S_3,A_3$

$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$

$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$

What results do we have for higher orders and are there any results for a general $n$?

I think the answers for the first few degrees ($n$) are:

$n=2$, $S_2$

$n=3$, $S_3,A_3$

$n=4$, $S_4,A_4,D_4,\mathbb{Z}_4,K_4$ ($K_4$ is the Klein four group)

$n=5$, $S_5,A_5,D_5,\mathbb{Z}_5,Fr_5$ ($Fr_5$ is a Frobenius group)

What results do we have for higher orders and are there any results for a general $n$?