Suppose you have a relative elliptic curves $f : E\rightarrow S$ (say $S$ is connected). Then suppose you have two sections $g,g' : S\rightarrow E$, corresponding to two sections $g_*,g'_*$ to the map of fundamental groups $$\pi_1(E)\stackrel{f_*}{\longrightarrow}\pi_1(S)$$ Technically, we have to choose a geometric point $s\in S$, and then the sections $g,g'$ would correspond to morphisms $g_*,g'_*$ to $\pi_1(E,g(s))$ and $\pi_1(E,g'(s))$. However, since these groups are canonically isomorphic up to inner automorphisms, it makes sense to ask, are $g_*$ and $g'_*$ conjugate?
I'm pretty sure there can generally be more than 1 conjugacy class of sections of $E/S$. Is there a nice way to understand these conjugacy classes, or to recognize when two sections are conjugate?
