Flight Instructor with student reported an NMAC in the traffic pattern when a military helicopter turned directly underneath the aircraft to fly to a nearby heliport. The Flight Instructor made an evasive maneuver to avoid the helicopter.

Narrative

My student and I were conducting night landings at ZZZ in Aircraft X for his PPL training and were flying a left hand pattern for Runway XX at 3;000 feet MSL and talking on CTAF frequency XXX.YY with our position lights; wingtip strobe lights; flashing red beacon and landing light on for visibility. As we were approaching short final a helicopter under the callsign XXX (Aircraft Y in this report) called that they were holding short of Runway XX and would depart after we conducted our stop and go. We landed; stopped; and then departed Runway XX and called Departure Leg XX and that we were staying in the pattern after we were airborne. The helicopter called that they were departing Runway XX. After we reached 2;600 feet MSL/700 feet AGL we called turning left crosswind for Runway XX. Almost immediately after our crosswind call I heard the helicopter call left crosswind as well and was able to turn my head and verify their position behind us on crosswind at roughly our same altitude. We then called turning downwind for Runway XX. Abeam the approach end of Runway XX and at 3;200 feet MSL we reduced power and started configuring the aircraft for landing. Approx. 1 mile north of the airport and 2;900 feet MSL we cleared the base leg and final leg of the pattern for traffic and turned left base for Runway XX while making our position report on the radio. About the time we finished our base turn and were on a heading of approximately 210 we noticed the red beacon light of the helicopter approaching quickly from our left approximately 200 feet in front of us and 100 feet below us. I executed an immediate full power climb to avoid the helicopter; climbing back up to 3;200 feet MSL. At this time the helicopter called that they were turning left base for Runway XX with no indication that they had us in sight. Since our aircraft was in a nose high position we had lost sight of the helicopter and so I requested their position at the same time as announcing our actions on CTAF. They responded that they were descending to a final turn for the heliport that is located just east of the approach of Runway XX. I lowered the nose of our aircraft; got the helicopter in sight and stabilized our approach to Runway XX. Both aircraft landed with no further incidents. Had the helicopter asked us to extend our downwind so they could turn base inside of us we would have been happy to do so as we had already extended our downwind for them once before when they approached the airport for the first time on a straight in approach for Runway XX. I understand that their helicopters are fast and can fly shorter approaches than our fixed wing aircraft but without proper communication their actions are likely to cause a midair collision in the future. Per 14 CFR § 91.113.f: Each aircraft that is being overtaken has the right-of-way and each pilot of an overtaking aircraft shall alter course to the right to pass well clear. We were the aircraft being overtaken so we had right of way but the helicopter did not follow proper procedure and proceeded to overtake us on the left hand side without any communication causing us to turn left base into them. Also per 14 CFR § 91.113.g: When two or more aircraft are approaching an airport for the purpose of landing; the aircraft at the lower altitude has the right-of-way; but it shall not take advantage of this rule to cut in front of another which is on final approach to land or to overtake that aircraft. While the helicopter was approximately 100 feet below us; it was most definitely trying to overtake our aircraft and cut in front for final. Had I not taken immediate evasive action there was a high possibility of a midair collision that was completely avoidable if the helicopter had been communicating its intentions correctly and not operating in a careless and reckless manner by violating 14 CFR § 91.113.f and 14 CFR § 91.113.g and 14 CFR § 91.13.a.