Suppose that $s_1 \neq s_2$ are in $S$ and $f(s_1) = s_2$, where $f$ is an element of a permutation group. Then if $H = \{f \in A(S) \mid f(s_1) = s_1\}$ and $K = \{g\in A(S) \mid g(s_2) = s_2\}$, we need to show that:
1) If $h \in H$ then there exists $g \in K$ such that $h = f^{-1}gf$.
I can't see any connection between $h \in$ H and existence of $g$.
One possible way to show that something exists is to explicitly construct it. Think of the requirement $h=f^{-1}gf$ as an equation and solve it for $g$. You should get that $g=fhf^{-1}$. Now show that this $g$ is what we want, i.e. prove that this $g$ satisfies the given requirements: that $h=f^{-1}gf$ (which is true simply from how we defined this $g$) and $g\in K$. For the latter, you will need to use what you know about $f$ and $h$.