What is the term for directly overseeing and inspecting an activity?

The term that best fits the definition of directly overseeing and inspecting an activity is "supervision." Supervision involves the management of a specific process or a group of individuals by ensuring that tasks are being completed effectively and according to established guidelines. It requires a hands-on approach where the supervisor monitors performance, provides guidance, and ensures that objectives are being met in real-time. This level of direct involvement is essential for facilitating communication, addressing issues as they arise, and fostering a productive working environment.

In contrast, management encompasses a broader range of activities, including planning, organizing, and coordinating resources, rather than the direct oversight implied by supervision. Administration typically refers to the implementation of policies and procedures within an organization and is not confined to the direct oversight of a specific activity. Oversight, while related, generally implies a broader view or a higher-level monitoring rather than the hands-on inspection and immediate feedback that supervision entails. Therefore, "supervision" is the most fitting term for the specified activity of overseeing and inspecting directly.