There are many aircraft operations that require the use of task specialists. These are crew members who do not fly the aircraft, but instead operate equipment in order to perform the mission.

For example, pollution patrol flights are typically flown with task specialists who operate radars, cameras and other sensors to detect the pollution, determine it's quantity and characteristics and then to record evidence if possible of any ship or gas / oil rig that is discharging.

If the task specialist detects an incident on board, they must report this to the pilot. Although flying the aircraft to allow the specialists to perform their duties, the pilot still has the ultimate safety responsibility. More information from EASA AIR OPS is here...

EASA AIR OPS
SPO.GEN.106 Task specialists responsibilities

(a) The task specialist shall be responsible for the proper execution of his/her duties. Task specialists’ duties shall be specified in the SOP.
(b) During critical phases of the flight or whenever deemed necessary by the pilot-in-command in the interest of safety, the task specialist shall be restrained at his/her assigned station, unless otherwise specified in the SOP.
(c) The task specialist shall ensure that he/she is restrained when carrying out specialised tasks with external doors opened or removed.
(d) The task specialist shall report to the pilot-in-command:

(1) any fault, failure, malfunction or defect, which he/she believes may affect the airworthiness or safe operation of the aircraft, including emergency systems; and
(2) any incident that was endangering, or could endanger, the safety of the operation.