On the meaning of "Class of finite groups". What do we mean precisely when we speak about a class of finite groups? Is this simply a collection of some finite groups, maybe collected with a criterion (example: the class of all finite cyclic groups), or there exists a precise definition of "class of finite groups" involving some other properties?
In the case a class of finite groups is simply a collection of finite groups which follows some criterion (ex finite cyclic groups as above), what can we say about a collection of some finite groups collected randomly? Ex $\{S_5, A_3,D_4, C_9\}$... I'd call it simply a set of finite groups. Is this right?
Thank you all

The standard definition of a class of groups is a a class $\mathcal{C}$ (in the set-theoretic sense) whose members are groups with the property that if $G$ and $H$ are isomorphic groups then  $G \in \mathcal{C} \Leftrightarrow H \in \mathcal{C}$.
So yes, there is a class of finite cyclic groups and a class of nilpotent groups, etc. It is customary to study the closure properties of specific classes. For example, a class may or may not be closed under subgroups, quotient groups, direct products, free products, etc.
