Groups like symmetric group

Question

For sufficiently large n consider this question Let $G$ be a finite group with the following properties:

1. $|G|=n!$

2. $H,K$ are subgroups of $G$ such that $H\cap K=1$ and $H\cong S_{3}$ and $K\cong S_{n-3}$

3. $t$ is an involution in $H$ such that $|C_G(t)|=2(n-2)!$

Now, is it true that $C_G(t)\cong \langle t \rangle \times S_{n-2}$

Answer 1

How about $D_{n(n-1)}\times C_{n-2}\times S_{n-3}$ for any $n$ where 3 divides $n(n-1)$ but 4 doesn't?