Are two finite groups of the same order always isomorphic? Are two finite groups of the same order always isomorphic? Some simple example would be great!

I don't know if you know the group action but, you can review this counterexample for the futhure. Consider the following groups:
$$G_1=\{\text{id},(1,2),(3,4),(1,2)(3,4)\}\\ G_2=\{\text{id},(1,3)(2,4),(1,4)(2,3),(1,2)(3,4)\}$$
Although, both are isomorphic to $\mathbf{V}$ of order $4$; they are not isomorphic of permutation groups point of view. In fact, it easy to check that $(G_2\mid\Omega)$ transitively while $(G_1\mid\Omega)$ is not wherein $\Omega=\{1,2,3,4\}$.
