A Flight Instructor in a C172 reported an NMAC in the pattern of a tower controlled airport.

Narrative

My student and I were flying a Cessna 172S under VFR conditions at ZZZ at the conclusion of a routine training flight from ZZZ1. We had been cleared to enter; extend our downwind of ZZZ and to remain west and clear of ZZZ2's airspace; which is a known standard practice here which involves extending downwind and staying west of ZZZ2's Departure corridor. While on the extended downwind; we observed a Baron in the pattern for the same runway (XX) on an opposite downwind. The Baron initiated a turn to base while I was still well on my extended downwind leg. This created a potential head-on course; as I could see the Baron's flight path aligning with mine.At this point; ZZZ Tower had not provided any acknowledgment or information about the Baron; nor had I been instructed to follow this aircraft as my traffic to follow. With the assumption that we were on a collision course; I initiated a hard right turn to avoid the Baron. The resulting maneuver likely appeared as though I cut the Baron off; but upon reviewing our ground track; the Baron's turn to final did not seem to align with ZZZ's RWY XX heading; which further clarified the situation. My student and I banked and broke off to the right to avoid their path. It was only at this point when we were close to the Baron; that we were told this was our traffic to follow. After completing the maneuver; I continued to monitor the traffic in the area and we were able to maintain safe separation.Factors Contributing to the Near Miss. Tower Task Saturation: The heavy traffic load at ZZZ; combined with limited communication regarding the Baron's location and intentions; led to confusion and a lack of situational awareness. There were 8 planes on Taxiway 1 and 4 additional planes in the pattern. Baron Overshooting the Base to Final Turn: The Baron may have overshot their base to final turn; creating an unexpected traffic conflict for my student pilot and I while we were still on our extended downwind. ADS-B outage; the aircraft had intermittent ADS-B In connectivity and this contributed to a source of our confusion.Evasive Maneuver Description. Upon recognizing the potential for a collision with the Baron; my Student Pilot and I:executed a hard right turn to increase separation. Communicated my actions on the ZZZ Tower Frequency. Monitored any other aircraft in the pattern to ensure continued safe separation.Lessons Learned. This event reinforced the need for clear communication with ATC; especially during periods of high traffic; to maintain awareness of all relevant traffic in the area. We also learned the importance of quick thinking and remaining calm during a potential conflict in the pattern. Recommendations. Tower Task Saturation Management. ATC should be aware of traffic saturation and ensure they provide adequate communication and traffic advisories; especially during peak times like weekends at ZZZ. Overshoot of Base to Final Turn. Pilots; particularly in congested airspace; should be mindful of maintaining proper alignment with their final approach course to avoid unanticipated conflicts.