The "philosophy" of (group) actions is such that the most rigid conditions we can think of (abstractly) are freeness and transitivity.
Let me introduce some terminology to make this idea clearer. For $G$ a group, we call a set $X$ a rigid set for $G$ if there exists a group action $. :G \times X \rightarrow X$ that is both free and transitive.
Does any group $G$ admits a rigid set $X$? What can we say about the category of rigid sets for $G$ where morphisms are $G$-equivariant maps?
PS: this question is motivated by the fact that in practice, most natural actions aren't free but only faithful. For example, the natural action of $SO(3)$ on $\mathbb{R}^3$ is faithful and transitive, but certainly not free. Yet, it seems to me that the "good object" on which a group $G$ should act has to be a rigid set for $G$. Does a rigid set for $SO(3)$ is known to exist ?