Helicopter Flight Instructor reported while taking off from a stop and go another aircraft flew directly over them; resulting in both aircraft taking evasive action to avoid a collision.

Narrative

While conducting Flight Instruction with a brand-new Private Pilot Helicopter Student; we were executing closed traffic patterns at an uncontrolled local airport; heavily used by multiple flight schools in the area. Prior entering the airport environment; I monitored the Common Traffic Advisory Frequency (CTAF); visually scanned & identified one aircraft in the airport's standard Left Pattern; utilizing the anticipated runway with light prevailing winds (Runway XX) on the east side of the airfield. As a helicopter; I crossed over to west side of the airfield; to enter a mirrored traffic pattern utilizing right turns to RWY XX; in order to avoid the airplane pattern flow; outside of final and departure paths to RWY XX. As this was an initial introduction to flying in the airport environment we will be using often; the intention of training was demonstrating the rectangular pattern; announcing actions / intentions; clearing for traffic & integrating with other aircraft at an uncontrolled airport with timing; altitude separation & making / monitoring radio position reports. After completing one normal traffic pattern; announcing & conducting a full-stop to the mid-field of RWY XX at the intersection of RWY XY/XZ; I completed my pre-take off checks & announced departure from RWY XX to remain in right closed traffic again over the CTAF. I cleared verbally & visually as a standard practice left; right & above before initiating a normal open-area / accelerative (airspeed over altitude) take-off profile. No other radio call was heard by either my student or myself; either prior to our departure call nor after our take-off to indicate there was any other airplane on final approach behind us; as we were still occupying RWY XX; until our departure leg radio call from mid-field RWY XX. Shortly after take-off & just prior to my right turn onto the Cross-Wind leg of our western pattern; I verbally & visually announced again my clearing left; center; right as I was about to then make the right Cross-Wind CTAF call...at that same moment I was clearing right to begin my turn; I noticed 'TRAFFIC +100 feet' in bold yellow on my GPS display out of my peripheral vision. *(I was sitting left seat; looking / turning right with the GPS located center / low on our console between the pilot seats). The traffic warning (ADS-B) display appeared in that instant to indicate the traffic was +100 feet above & in front on me; however I had just cleared left; center; right & above in our climb to Cross-Wind. I instinctively processed they have to be immediately behind us & increased our right turn maneuver onto Right Cross-Wind; in order to clear the presumed airplane path on departure from RWY XX; which I believed should at least procedurally continue straight out or turn away to Left Cross-wind; alternatively. Upon my right turn; I visually acquired the Light Sport Airplane appearing almost on our tail out the right side doorway (doors were removed). The LSA just then began maneuvering away to their left from my perspective; after nearly flying directly into us from behind while climbing through I estimate 200 feet Above Ground Level (AGL) & just beyond the departure end of RWY XX. I completed making the right Cross-Wind radio call on CTAF; announcing I also had the airplane directly behind us in sight. We proceeded around the pattern; completing another normal approach; then decided to depart the pattern thereafter to avoid any further airplane interference concerns; as it left my student visibly shaken; understandably so. A Solo Student Pilot was apparently the other aircraft involved in the NMAC with us. My radio calls were clear & repetitive enough that my tail number was recalled; however I did not recall hearing the student pilot's radio calls or tail number during their pattern operations; however. We engaged in a very productive & detailed retelling of events; from the LSA Student Pilot's viewpoint as well as mine.There could have been radio transmission issues from their LSA; as I did not recall hearing their pattern calls; outside of a remark the LSA made after the NMAC over CTAF; directed towards me. The LSA saw us on the runway when they made their final approach; deciding to execute a 'Go-Around'; which seemed to have occurred as I made my departure take-off call & climb out concurrently. We decided it could have been their 'Go-Around' radio call on CTAF either did not transmit; or was made simultaneously with my departure CTAF call perhaps; stepping on both our intended transmissions. They did not hear my departure call & we never heard a final or Go-Around call it seemed. Regardless; being an uncontrolled / Class G Surface airport; no radios or calls are technically required; so 'See & Avoid' responsibility prevails for each pilot operating there. The concern overall for the submission of this report; is the LSA pilot's decision to initiate a final approach with the runway occupied by a stationary helicopter; followed by the appropriate decision to abort the landing & 'Go-Around'; however followed directly by continuing on their flight path down the runway; actually deviating to the right initially (to THEIR 'upwind' side; as they were taught (primacy)- not understanding or recalling the path towards me was in direct conflict with my right turns /right traffic patterns). Only after nearly colliding; did the LSA with the student pilot appear to maneuver to deviate away from us - again; from behind; with us in sight at a higher closure speed & flight path presenting a direct collision hazard - which was entirely & physically invisible & unseen to us in the helicopter. I believe the impetus for avoiding the NMAC was the responsibility of the LSA pilot in that specific position - from their decision to continue on approach with an occupied runway; abort; Go-Around & their Climb-out path - with us in sight the entire time. They were never in a viewable position to myself or my student at any point from the completion of our final approach; full-stop & before though take off procedures with CTAF calls. They possibly & likely experienced a 'freeze' reaction under stress as a student pilot; resulting in the inability to determine an alternate course of action when the initial 'Go-Around' & Up-wind deviation decision didn't resolve the avoidable traffic conflict. Working together to improve Situational Awareness; maintaining Vigilance; understanding mixed aircraft type operational environments; using all available resources for traffic avoidance & deconfliction (fully operational radios; ADS-B Alerts; Full-Aircraft Lighting; etc) & following established procedures will provide the best practices for the 'See & Avoid' visual & dynamic training environment present here.