= Solution
Two <group actions> on $S_1,S_2$ are equivalent if there is a <bijection> $\phi:S_1\to S_2$ such that $\phi(gs)=g\phi(s)$ for all $g,s$. Such a bijection is <equivariant>.
The <orbit-stabilizer theorem> in this form says that the action on $O_s$ is equivalent to the left <coset> action on $G/G_s$, by $gG_s\mapsto gs$. This map is well-defined and injective because $gs=hs$ exactly when $h^{-1}g\in G_s$, and it is onto by the definition of the <orbit>. It is equivariant by multiplication.
Thus every transitive action is a <coset> action. Two such actions $G/H$ and $G/K$ are equivalent exactly when $H,K$ are conjugate: <stabilizers> at different points of one <orbit> are conjugate, and a conjugacy explicitly supplies a coset-action equivalence. This last classification is what makes counting by <orbit> types possible.
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