Recently I was directed to the topic of minimal bases for permutations acting on finite sets via a question about stabilizer subgroup chains by Dr. Holt. While making my way through some readings on the topic, I noticed the restriction of the permutation group $G$ must always act transitively on a set $\Omega$. I wondered what the complications from a intransitive group would be so I decided to look further.
I have read here that "... For an intransitive group $ ( G, X) $ with orbits $ X _ {i} $,
$$ X = X _ {1} \cup \dots \cup X _ {s} , $$
and the restriction of the group action to $ X _ {i} $ is transitive."
This makes sense, and my intuition tells me that a minimal base for $X$ would consist of $\sum_{i=1}^sb(X_i)$, since $G$ transitively acting on $b(X_i)$ would generate all elements of that particular orbit $X_i$, but I admit this is just a hunch. Here $b(X_i)$ denotes the minimum base for each $X_i$
However, it continues on to say that "Let $ H $ be a subgroup of a group $ G $ and let
$$ G = H \cup Hx _ {1} \cup \dots \cup Hx _ {s - 1 } $$
be the decomposition of $ G $ into right cosets with respect to $ H $. Further, let $ X = \{ Hx _ {i} \} $ (Not sure I understand this notation). Then the action of $ ( G, X) $ is defined by $ ( Hx _ {i} ) ^ {g} = Hx _ {i} g $. This action is transitive and, conversely, every transitive action is of the above type for a suitable subgroup $ H $ of $ G $."
Question: If I am correctly understanding this, then by choosing an appropriate subgroup $H \leq G$ and applying $\forall g\in G$ to the right cosets of $H$ we can obtain a transitive action for all $X$. Does this mean that we can then take the steps to determine a minimal base for this transitive action? What is the criterion for choosing the appropriate subgroup $H$?
Context:"Bailey, P. (2011). Base size, metric dimension and other invariants of groups and graphs. Bulletin of the London Mathematical Society, 43" makes a clear and direct relationship between minimal bases of permutations acting transitively on a set and what is referred to as the determining number of graph. I understand the connection for a graph that is vertex-transitive, but how can we handle the cases for a graph automorphism group that has multiple orbits?
Full disclosure: I'm an undergrad who has likely muddied up some notation, conventions, and operations. I warmly welcome feedback.