What does it mean for an action to be isomorphic to another action?

Question

In Aluffi Algebra Chapter 0, proposition 9.9 in chapter 2, there is the following statement:

Every transitive left action of a group $G$ on a set $A$ is isomorphic to the left multiplication of $G$ on $G/H$, for $H$ the stabilizer of any $a\in A$.

My question is, what does isomorphism in this sense mean?

Answer 1

Let $G$ acts on the sets $A$ and $B$. We say that these actions are equivalent (isomorphic) if there is a bijection $\phi$ from $A$ to $B$ such that $$g.\phi(a)=\phi(g.a) \text{for all } g\in G \text{ and for all } a\in A.$$