MongoDB Aggregate groupBy date and count on subdocs

I know theres other versions of this question on here, but I'm struggling to get this to line up. I have docs that are similar to:

[
{_id: 1, date: "2022-04-08T23:30:12.000Z", books: [{author: "Johnson", title: "First Title"}, {author: "Smith", title: "Second Title}]},
{_id: 2, date: "2022-04-22T23:30:12.000Z", books: [{author: "Johnson", title: "Some Other Title"}]},
{_id: 3, date: "2022-05-05T23:30:12.000Z", books: [{author: "Smith", title: "Title Round 2"}]},
{_id: 4, date: "2022-05-15T23:30:12.000Z", books: [{author: "Johnson", title: "Found a Title", {author: "Smith", title: "Wrote again"}, {author: "Brooks", title: "New Title"}]}
]

I'm trying to group the documents by month-year and then run a count on the times a distinct value shows on the author field. So far I have a pipeline that looks like:

{
          "$unwind": "$books"
        },
        {
          $project: {
            _id: 1,
            books: 1,
            month: {
              "$month": "$date"
            },
            year: {
              "$year": "$date"
            }
          }
        },
        {
          $project: {
            _id: 1,
            books: 1,
            date: {
              $concat: [
                {
                  $substr: [
                    "$year",
                    0,
                    4
                  ]
                },
                "-",
                {
                  $substr: [
                    "$month",
                    0,
                    2
                  ]
                },
                
              ]
            }
          }
        },
        {
          $group: {
            _id: {
              date: "$date",
              books: {
                freq: {
                  $sum: 1
                }
              }
            }
          }
        },
        {
          $project: {
            "_id": 1,
            "date": 1,
            "books": 1
          }
        },
        
      ]
    }

I aiming for a final output that looks like:

[
{date: "2022-04", authors: { "Johnson": 2, "Smith": 1}},
{date: "2022-05", authors: {"Johnson": 1, "Smith": 2, "Brooks": 1}}
]

I've seen ways to run a count on the subdocs, but in trying to implement I'm losing my group by date or just getting errors. I've seen enough to know its doable, just lost trying to get it just right. Any help is appreciated.

your first 3 stages can stay the same (assuming the dates are stored as date objects and not as strings)

after that

1. group by $date and $books.author fields and count the occurrence of each group. This will give the counts you need in the final answer
2. Then group by $date only and push each count to an authors array in the format of key value {k:key,v:value} so that it can be converted to an object in next stage
3. $arrayToObject on authors array to convert it to an object

if you also want to sort the date add a { $sort: {date: 1 } } stage

db.collection.aggregate([
  { $unwind: "$books" },
  { $project: { _id: 1, books: 1, month: { "$month": "$date" }, year: { "$year": "$date" } } },
  { $project: { _id: 1, books: 1, date: { $concat: [ { $substr: [ "$year", 0, 4 ] }, "-", { $substr: [ "$month", 0, 2 ] } ] } } },
  {
    $group: {
      _id: { date: "$date", author: "$books.author" },
      count: { $sum: 1 }
    }
  },
  {
    $group: {
      _id: "$_id.date",
      authors: { $push: { k: "$_id.author", v: "$count" } }
    }
  },
  {
    $project: {
      _id: 0,
      date: "$_id",
      authors: { $arrayToObject: "$authors" }
    }
  }
])

playground you can look at the intermediate results from the Stage dropdown on top