I know that $A_4$ is a transitive subgroup of $S_6$, so transitivity is not critical in this problem.
I tried to find irreducible polynomials of form $X^6-a(a \in \mathbb{Q})$, but their Galois groups seems to be $D_6$ when the order is $12$.
Please let me know if there is a simple example. I think there is no degree $6$ irreducible polnomial $f(x) \in \mathbb{Q}[x]$ whose Galois group is $A_4$.