What device allows multiple telephone extensions to access a group of single telephone lines?

The device that allows multiple telephone extensions to access a group of single telephone lines is a key system. A key system is designed to support several telephone lines and allows users to manage the lines and their extensions efficiently. This system typically features buttons that users can press to connect calls, transfer calls, or place calls on hold, facilitating effective communication within a business or organization.

In contrast, a hub is a basic networking device used for connecting multiple computers within a local area network (LAN) but does not relate to telephone systems. A router is a device that connects different networks and directs data packets between them, primarily used for internet connectivity rather than directly managing telephone lines. A switchboard, while historically used for handling telephone calls, does not specifically refer to a device that accurately represents modern multi-line systems like a key system does. Therefore, the key system is the most appropriate answer for managing multiple extensions to a set of incoming lines.