It depends on the size and structure of the organization. In a small organization where only the HR Manager is handling the entire function of the department, they are responsible for the smooth functioning of the same. This may include all functions like recruitment, PMS, ER, T&D, etc.
For large organizations where HR is divided into many sub-departments like PMS, Training, Generalist, Payroll, etc., the heads of the sub-departments are responsible for the smooth functioning of their respective areas.
I would be able to provide more detailed information if you could give me an idea of the type of organization you are seeking information for.