This is not easy to answer in a short paragraph. But basically what you want to do is use binding for your service.
It's certainly not a trivial task, but it's the elegant way to go. Besides of initiating your service with startService()
you would additionally bind to your service. The binding mechanism allows your Activity to have a pointer to your Service class. With this pointer you can do whatever you want, including passing a pointer of the Activity itself to the Service, so the service would have a pointer to your Activity.
Once the Service has a pointer to your Activity, you can call any methods and set any variables you want, including setting the return value from your AsyncTask.