GA flight instructor with a student reported an NMAC during climbout from Runway 22L at JRF airport requiring evasive action. Reportedly; the conflicting aircraft was NORDO and on approach to Runway 4R.

Narrative

I was conducting a training flight and was working takeoffs and landings with a student on Runway 22L out of JRF. The winds had become a tail wind so after doing a touch and go Tower instructed us on our initial climbout to fly straight out; make a tear drop entry and report established on a straight in final for Runway 4R. As we were climbing to traffic pattern altitude (800 feet MSL) I noticed some lights that I perceived to be in the distance at our 12 o'clock and at a higher altitude. Initially I thought it might be approaching traffic for 8L into PHNL as overhead traffic is a regular occurrence. As I focused in on the lights I noticed the lights descending much more rapidly than I anticipated and it became clear to me that I was on a head on course with a plane descending into JRF for Runway 4R. I took the controls and immediately made a climbing right turn to avoid traffic while simultaneously contacting Tower to ask if they were in contact with an aircraft making a straight in for Runway 4R. Tower initially said no then immediately recognized the aircraft we deviated for and called to the aircraft to contact them. The NORDO aircraft immediately responded with apologies that they never received a hand-off so they never made contact despite entering PHJR's Class D airspace. The aircraft then went around; entered right traffic for Runway 4R; and later made a downwind departure to exit the Class D to the west. We circled back to make straight in for Runway 4R to continue the flight.