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?