How do I update an embedded document with a nested hierarchy?

I have an embedded document that needs to be updated. I have managed to get everything working except for one scenario: I am unable to get an embedded document with a nested hierarchy to get updated. Here is an example of what I'm attempting to do:

console.log('Phone type: ' + req.body.phone.type); // prints, e.g., 'Phone type: Work'
console.log('Phone #: ' + req.body.phone.number); // prints, e.g., 'Phone #: 555-555-5555'

var updateData = {
        "user.$.contact_info": {
            email: req.body.email,
            phone: {
                type: req.body.phone.type,
                number: req.body.phone.number
            }
        }
    };
    Group.update(
        { "user._id" : req.params.user_id },
        { $push : updateData },
        function(err,data) {
            console.log('Success!'); // I see the "Success!" message every time I save a record
        }
    );

Schema:

var contactInfoSchema = mongoose.Schema({
    created : {
        type: Date,
        default: Date.now
    },
    email : String
    phone: {
        type: String,
        number: String
    }
});

var userSchema = mongoose.Schema({
    created : {
        type: Date,
        default: Date.now
    },
    contact_info : [contactInfoSchema]
});

var GroupSchema = mongoose.Schema({
    created : {
        type: Date,
        default: Date.now
    },
    users : [userSchema]
});

What I find is that I can create records, but only the email address is stored, not the phone information. When inspecting the console I can see that the phone type and phone number info is being sent, but it just isn't updating in the database. What am I doing wrong?