Equivalence of definitions of conjugate subgroups

Statement
Let $$G$$ be a group and $$H,K$$ be subgroups. Then, the following are equivalent:


 * 1) $$H$$ and $$K$$ are conjugate subgroups in $$G$$: there exists $$g \in G$$ such that $$K = gHg^{-1}$$
 * 2) There exists a transitive group action of $$G$$ on a nonempty set $$S$$, and (possibly identical) points $$s,t \in S$$ such that $$H,K$$ are the isotropy subgroups at $$s$$ and $$t$$ respectively.
 * 3) There exists a group action of $$G$$ on a set $$S$$, and (possibly identical) points $$s,t \in S$$ that are in the same orbit under the $$G$$-action, such that $$H,K$$ are the isotropy subgroups at $$H$$ and $$K$$.