What is meant by group action are isomorphic?

Question

Let $\pi_1 : G \times S \mapsto S$ is one group action and $\pi_2 : G^{'} \times S \mapsto S$ is another group action. What is it mean by the statement that $\pi_1 \cong \pi_2$?

Answer 1

It means that there is a bijection $f\colon S\to S$ which is also a morphism of $G$-sets, a $G$-equivariant map, i.e. that $f(gs) = gf(s)$ for all $s\in S,g\in G.$