I think so. This might be the right definition of the group being maximal. But I'm looking for an algorithm I can turn into a computer program. How could I possibly find the set of all invertible matrices $A,B$ s.t. $MA = BM$?

@NicolasTholozan I want quotients induced by a Lie group action. One of the situations these quotients occur is when you have a Lie group $G$ and a normal Lie subgroup $H$, then $H$ acts on $G$ giving a quotient manifold $G/H$ (if the action is free and proper). But I care about other group actions too, not those of Lie subgroups.

I looked at the limit for scalars but I couldn't solve it either. I can unroll the $(1- 1/n bdb)^k$ term into a binomial expansion but I don't know how I would turn that expression into an exponential.

@მამუკაჯიბლაძე thank you very much. This is a much cleaner way to phrase the question.

Ok. I think I understand. At the singularity there isn't an open neighborhood that is mapped homeomorphically. What I'm I working with then? Is there a word for covering maps with singularities?

I thought it was. Every point in the image has a discrete fiber and the reflections are the automorphism group of the map. There are singularities so some fibers have different numbers of points, but I don't think that stops it from being a covering map.

$\mathbb R^n$ as a manifold. The functions $h_i$ would be embeddings or submersions in the linear case and covering maps if they are the absolute value.