The Executive Council (EC) is the highest formal instrument of government, through which the government collectively and formally advises the Governor-General.
All Ministers of the Crown are members of EC (whether those Ministers are inside or outside Cabinet). EC is presided over by the Governor-General, who is not a member of EC.
EC usually meets at 4pm every Monday after Cabinet, if there are items for consideration. At the meetings, Ministers give formal advice to the Governor-General to sign Orders in Council (to make, for example, regulations or appointments). The submission of almost all items for consideration by EC must be authorised by Cabinet.
For further information on EC, see chapter one of the Cabinet Manual.
The Governor-General’s website provides information on the role and functions of the Governor-General, and Government House.
If the Governor-General is overseas, then the Administrator of the Government fulfills the Governor-General's functions.
The Administrator of the Government is the Chief Justice of New Zealand.