Does isomorphism transfer the transitative property between permutation groups?

Question

If $G$ is isomorphic to $G'$ and $G$ is transitive to $S_n$ then does it not immediately follow that $G'$ is also transitive to $S_n$?

Do I need to state some results or theorems to prove this or is it just trivial?

Answer 1

No. Firstly there is no reason to suppose that $G'$ is a subgroup of $S_n$ at all. But even if we assume that, then the claim is not true.

$G = \langle (1,2,3,4,5,6) \rangle$ is a transitive subgroup of $S_6$, but $G' = \langle (1,2)(3,4,5) \rangle$ is intransitive. They are both cyclic groups of order $6$.

An even smaller example: $G = \langle (1,2)(3,4),(1,3)(2,4) \rangle$ is a transitive subgroup of $S_4$, but $G' = \langle (1,2),(3,4) \rangle$ is intransitive. Here $G \cong G' \cong C_2 \times C_2$.