Constructing metric tensor of cosmological models viewed as (Riemann) homogeneous space and connections with the Killing vectors

Question

As we know cosmological models are Riemann manifolds which are assumed to have some sort of symmetries (spherical, isotropic, homogeneity and etc.) and the problem is to find the form of the metric tensor which obeys these symmetries. Using the group of isometries over manifold $I(M)$ (or some of its subgroup) that acts transitively as a group action over $M$ we can express $M$ as a orbit manifold (for example $N\cong SO(3)/SO(2)$ in the case of spherical symmetry with $SO(2)$ being stabilizer of some point and $N$ orbit of $M$). One of my questions is how to unify the metric tensor when once found its form on the orbit? The next thing that bothers me is the classification of the cosmological models using a group action and Killing vectors. In some resources they say that $G_{3}$ (which I cannot find what is) is the group of isometries and start using some Killing vectors (without mentioning where they come from). So what are these $G_{3}$, $G_{2}$ groups and why do not they use these groups to work on the orbits and the unify the metric tensor over $M$?