If two permutation groups on the same set are isomorphic are they the same?

Question

For any permutation groups on a set $S$ say $G,H\subseteq \text{Sym}(S)$ is it true that $G\cong H\implies G=H$? My gut says its trivially false or trivially true i.e. this is likey a dumb question, but I'm confused.

Answer 1

In $S_6$, the group generated by the cycle $(123456)$ is cyclic of order 6, and the group generated by $(12)(345)$ is cyclic of order 6. But, they are not the same. This example shows also that they may not even be conjugate subgroups.

Answer 2

No, this isn't true. Consider $S_5$ (not minimal but easy). Then consider the groups $\langle (1\,2)\rangle$ and $\langle (3\, 4)\rangle$. Both are isomorphic to $\mathbb{Z}/2\mathbb{Z}$, but they're not equal.