I have not been able to find a proper definition of what an isomorphism class is (in the context of group theory). If one could define it properly for me and give me some help with the following two questions, it will be deeply apprecited:
1) Let $p$ a prime and $X$ a set of cardinality $p+1$. How many isomorphism classes are there of $G-sets$ where $G =\mathbb{Z}/p^2\mathbb{Z}$. What if $G =\mathbb{Z}/p\mathbb{Z} \times \mathbb{Z}/p\mathbb{Z}$?
2) List (without repetitions) the isomorphism classes of all transitive actions of $S_4$ on a set of cardinality 3 or 4.