Customer Communication via Customer Portal

Currently it is possible to communicate with customers with the Help Desk feature. Customers who have access to the customer portals can see the ticket in the customer portal and chat with our support team. BUT the customers also always get an E-Mail.
It would be great to have the opportunity to turn this E-Mailing feature off. Because sometimes you want something to stay on the Customer Portal and not anywhere else.
Even better: Enable/Disable E-Mail communication based on the pipeline.