Final answer:
The Board of Health is the group that delegates its duties to the local health department, overseeing various public health services and programs.
Step-by-step explanation:
The group that delegates its duties to the local health department is the Board of Health. The Board of Health is typically a governmental organization at the state or local level with the responsibility to protect and promote the health of its community, which includes delegating certain public health responsibilities to local health departments. These departments then carry out a variety of services, including inspection of restaurants, vaccination programs, and response to public health emergencies.