What is the relation between a group action and its restriction to a normal subgroup?

Question

Let $G$ be a group operating on a set $E$ on the right. Let $H$ be a normal subgroup of G.

Let $E/G$ (resp. $E/H$) denote the quotient set of E under the relation of conjugation w.r.t the operation of G (resp. H).

Then:

1. $G$ operates on the set $E/H$ on the right: $g\mapsto(x.H\mapsto (x.H).g=(x.g).H)$;
2. $G/H$ operates on $E/H$ on the right: $gH\mapsto(x.H\mapsto (x.H).(gH)=(x.g).H)$.

Since the relation of conjugation w.r.t $H$ is finer than that w.r.t $G$, we have a canonical surjection $\phi:E/H\rightarrow E/G$.

Question: For $x\in E$, what can we say about the inverse image $\phi^{-1}(\{x.G\})?$