**Background**

This can be generalised, but let me be fairly concrete. Let $X$ be a simply-connected riemannian manifold and let $G$ denote the Lie group of isometries, assumed nontrivial. Let $F < G$ be a finite subgroup acting freely and consider the smooth quotient $X/F$ with the induced riemannian structure.

The normaliser $N(F) < G$ still acts on $X/F$ isometrically with $F < N(F)$ acting trivially. So $X/F$ inherits an isometric action of the group $N(F)/F$.

Now let $E < N(F)/F$ be a finite subgroup acting freely on $X/F$ and consider the quotient $(X/F)/E$. This is a smooth manifold, locally isometric to $X$ and hence isometric to $X/D$ for some freely-acting subgroup $D<G$.

**Question**

How is $D$ related to $E$ and $F$? I would expect $D$ to be an extension of $E$ by $F$. Is it? And if so, but does it split? And if not, is there a name for this construction?