I did not write that you should not have 2 documents. I wrote that one document should not refer to another document using a field that is mutable, dynamic, modifiable. Since the field groupName can be changed is not a good choice to maintain a relation from collection 1 to collection 2. Since _id cannot be changed, is unique and is indexed. As you see making the relation field modifiable it causes update difficulties.
updateMany works within a single collection. What you want is to change groupName in 1 collection to be reflected into another collection.
The problem of having a name change is not related to the way you get the name. It is related to the fact that your use-case allow to change. There is no issue in having groupName:group-a from one collection refer to another document in another collection using nameKey:group-a, the issue is
The problem is there because you want to change it and use it as a key. Yes show group-a is more user friendly than an _id, but users are not supposed to see your internal structure. Hopefully when you display a group your it name:“Group A” rather than it nameKey:“group-a”.