[MongoDB] How to add a field to a subdocument of every document in a collection?

Let’s say that a collection named items:

{
    "_id" : "a",
    "authorization" : {
        "group1" : "lrw",
        "group2" : "lr-"
    }
},
{
    "_id" : "b",
    "authorization" : {
        "group1" : "lrw",
        "group2" : "lr-"
    }
},
{
    "_id" : "c",
    "authorization" : {
        "group1" : "lrw",
        "group2" : "lr-"
    }
}

Now I would like to add a field to every “authorization”. The field may look like this:

{ "group3" : "lrw" }

Here’s the query that does that:

db.items.find({}).forEach(item => {
    db.items.update(
        { _id: item._id },
        { $set: { "authorization.group3": "lrw" } }
    )    
});