This is a very open question. By the way I'll try to explain the main concepts of a possible solution.
First of all, you need to consider that if you want real-time notification for "background-message" client B will need to keep connection open with socket (otherwise you'll need a sort of polling logic). Since B is connected to socket you can consider to work with Socket.io namespace in order to send this notification in another "room".
Another solution can be made by using Push Notification API provided by Firebase, AWS, etc.
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