GA instructor with student reported an aircraft in the pattern of LIC non-towered airport caused NMACs with multiple aircraft. Reporter stated they had to take evasive action to avoid a possible collision.

Narrative

I was in the traffic pattern at LIC with my student when I started to hear Aircraft Y talking with another plane over the CTAF. The aircraft was letting Aircraft Y know that they can't just change frequencies without Denver giving the okay they responded so clearly radios were working that they were having a hard time hearing them. The other aircraft was relaying what Denver wanted them to hear to squawk 1200 etc. They made a call that they were coming in to enter midfield left downwind for 34. They apparently did not see that there was multiple 5 including ourselves in the traffic pattern they came within 300 feet of us I called them on the CTAF and they responded so I know that their radios were working we were able to avoid them; then got very close to another Aircraft Z who was directly behind us and then came again within 200 feet of Aircraft A when they were on final and Aircraft Y then was 'in downwind' which was in final for 34. They made a call that they were on base but called it crosswind. We exited the pattern after they almost hit us and was departing the area since they clearly were not paying attention to anyone else. I called the club where they are training and spoke with the Director as well as with their CFI about the incident as well. They had 3 very close to being midairs back to back with no apparent realization or recognition.