**Edit: In my original framing of this question it was not so clear what I was looking for, so this is basically a re-write.**

I feel that I should already know the answer to this, but it never sits quite right in my head. 

When dealing with Gromov-Witten theory, one is obviously interested in part in the moduli stack of stable curves, $\overline{M}_{g,n}$ (actually, we probably should really care about the stack of *pre*-stable curves, but that's not going to be hugely relevant for my case).

When dealing with orbifold Gromov-Witten theory, we instead need to consider the moduli stack of *twisted* stable curves, which are those curves that may have isotropy at the marked points and nodes. For the problems that I am interested in, it is usually the case that every marked point has the same isotropy---in particular, this is $\mathbb{Z}/2$. Let us denote then the stack of twisted stable curves whose marked points all have $\mathbb{Z}/r$ isotropy as $\overline{M}_{g,n}^{\mathbb{Z}/r}$.

There is an obvious map $\overline{M}_{g,n}^{\mathbb{Z}/r} \to \overline{M}_{g,n}$ given by taking the coarse moduli space.

My question is: What is the relationship between these two stacks?