Light Transport; Low Wing; 2 Turbojet Eng
Communication Breakdown; Confusion; Situational Awareness
ATC Issue All Types; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance
Air Traffic Control Issued New Clearance; Air Traffic Control Issued Advisory / Alert; Flight Crew Became Reoriented; Flight Crew Requested ATC Assistance / Clarification; Flight Crew Returned To Clearance
Procedure; Airspace Structure; Chart Or Publication
After an exhausting morning we arrived [at] JFK for a passenger trip to ZZZ. The FMS would not accept the pre departure clearance so I did the clearance via radio with ATC. We were given the departure. I verified the clearance twice with ATC. The FMS was loaded both crew verified the clearance. After takeoff we were following the FMS for the departure; after CRI we were planning on the right turn to NEION intersection; that's when ATC questions our turn and he immediately assigned us a south bound heading. ATC gave us a telephone number to call after we arrive at ZZZ. We complied after landing at ZZZ. There was no threat to persons or property during this event.  ATC needs to be more specific during the clearance; he should have stated radar vectors after a heading off of CRI. Or there needs to be a waypoint after CRI. The climb and heading of is stated further down the SID but we missed it. (Possible fatigue onset) the heading needs to be more obvious. It was also communicated to the pilot in command by ATC personnel via the phone conversation that this departure has also been causing 'problems' for the airlines. This incident seems to happen once or twice a week says ATC. The crew fatigued upon reaching ZZZ.
An aircraft on a SID proceeded to a point on their flight plan after the SID transition. The aircraft are supposed to stay on the SID heading expecting radar vectors. This is not indicated clearly on the SID and causes confusion for aircraft on a regular basis.