gr.group theory - A question about iterated quotients in riemannian geometry

The group $D$ is the preimage of $E$ in $N(F)$, so it is as you expect. The finiteness hypothesis can be weakened, which is important for many applications. Things become clearer if one thinks categorically in terms of the universal properties.

Say an arbitrary group $G$ acts freely and properly discontinuously and isometrically on $X$ and $H$ is a normal subgroup of $G$. Then $G/H$ acts freely and properly discontinuously and isometrically on $X/H$ with $X rightarrow (X/H)(G/H)$ a $G$-invariant map. The induced map $f:X/G rightarrow (X/H)(G/H)$ is an isomorphism. Indeed, both sides composed back with the natural map from $X$ satisfy the same universal property, and $f$ respects the maps from $X$, so $f$ is an isomorphism. QED

In fact, with more work this can all be done more generally with $G$ and Lie group and $H$ a closed normal Lie subgroup, under suitable "niceness" hypotheses for the orbit maps (which are satisfied in the above situation): see Proposition 13 in section 1.6 of Chapter III of Bourbaki LIE.