How to aggregate data into quarterly buckets using MongoDB and Javascript?

Question

Could you guys please help me with the following query? I'm trying to modify the following piece of code for an application that I am working on.

  MongoClient.connect(url, function(err, db) {
      assert.equal(null, err);
      db.collection(collection).aggregate(
        { 
           $match : {Id: uniqueKey}
        },
        {  $unwind : '$' + docToUnwind },
        {   
            $project: {
                _id: 1,
                groupA: ('$' + docToUnwind + '.' + cashFlowField),
                groupB: ('$' + docToUnwind + '.' + cashFlowDateField),
                groupC: ('$Invoices.TotalAmt')
            }
        },
        {
            $match : { 
               groupB: { $lte: new Date(endDate),
                      $gte: new Date(beginDate) } 
            }
        },
        { 
            $project: {
                groupA: 1,
                groupC: 1,
                buckets: {
                   "yr" : {"$year" : "$groupB"},
                   "mo" : {"$month" : "$groupB"}
            },
         }
       },
       {
           $group: {
               _id: "$buckets",
               balance: {$sum:'$groupA'},
               total: {$sum:'$groupC'}
           }
       },
       function(err, result) {
           if (result) {
               callback(result);
           } else {
               callback(false);
           }  
        }
        );
       });

Output:

[ { _id: { yr: 2016, mo: 4 }, balance: 10, total: 10 },    
  { _id: { yr: 2016, mo: 11 }, balance: 20, total: 20 },  
  { _id: { yr: 2016, mo: 9 }, balance: 30, total: 30 },  
  { _id: { yr: 2016, mo: 7 }, balance: 40, total: 40 },  
  { _id: { yr: 2016, mo: 5 }, balance: 50, total: 50 },  
  { _id: { yr: 2016, mo: 6 }, balance: 60, total: 60 },  
  { _id: { yr: 2017, mo: 2 }, balance: 70, total: 70 } ]  

Is there a way that I can modify the above code to aggregate the above results into quarterly(or to be more generic - any frequency) buckets? The expected output that I am looking for is:

[ { _id: { qtr: Q1}, balance: 120, total: 120 },      
  { _id: { qtr: Q2}, balance: 70, total: 70 },    
  { _id: { qtr: Q3}, balance: 20, total: 20 },    
  { _id: { qtr: Q4}, balance: 70, total: 70 }] 

Additional Info:

yr: 2016, mo: 4, 5, 6 would fall into first bucket.
yr: 2016, mo: 7, 8, 9 would fall into second bucket.
yr: 2016, mo: 10, 11 and yr: 2017 mo: 0 would fall into third bucket.
yr: 2017, mo: 1, 2, 3 would fall into fourth bucket.

Sample document where,
cashFlowField = "Balance"; cashFlowDateField = "CreateDate";

    {
        "Id" : "2",
        "DocNumber" : "1002",
        "CreatedDate" : ISODate("2016-04-07T00:00:00.000Z"),
        "CustomerRef" : {
            "value" : "22",
            "name" : "EdgeLink, LLC"
        },
        "DueDate" : ISODate("2016-04-01T00:00:00.000Z"),
        "TotalAmt" : 1950,
        "Balance" : 1950,
        "MetaData" : {
            "CreateTime" : ISODate("2016-03-10T00:45:50.000Z"),
            "LastUpdatedTime" : ISODate("2016-03-10T00:45:50.000Z")
        }
    }, 

Edit: I have also tried in vain in implementing the following code using $in operator. I'm not sure what's going wrong in here.

db.collection.aggregate([
{$project:{"date":1,
       "quarter":{$cond:[{$in:[{$month:"$CreatedDate"},[4,5,6]},
                         "first",
                         {$cond:[{$in:[{$month:"$CreatedDate"},[7,8,9]},
                                 "second",
                             {$in:[{$lte:[{$month:"$CreatedDate"}, [10,11,0]},
                                         "third",
                                         "fourth"]}]}]}}},
{$group:{"_id":{"quarter":"$quarter"},"results":{$push:"$date"}}}
])

Thanks in advance for your help!:)