I know how fundamental group acts on fibers.
What does the following mean " ... constitutes an entire conjugacy class of subgroups of $G$ "
Let $S$ be a subgroup of $G$, then $cl(H)$ denote the conjugacy class of $H$. What is $H$ here?
We are taking set of all associated subgroups . This is a single set. I don't understand how this one set serves as conjugacy class of subgroups of $G$. For different subgroups we will have different conjugacy class.
Set of associated subgroup constitute entire conjugacy class of which subgroup of G.
This is not a single set as $\hat x$ is varying in $p^{-1}(x)$. The statement of the theorem is that the set $ \{ p_* \pi_1(\hat X, \hat x) \}_{\hat x \in p^{-1}(x)}$, which is a set of subgroup of $G$, form a conjugacy class, i.e is on the form $\{gHg^{-1} \mid g \in G\}$ for some subgroup $H \subset G$.