A TRACON Controller reported a NORDO aircraft flew below the Minimum Vectoring Altitude and into an adjacent facility's airspace. A previous sector failed to issue a frequency change to the aircraft resulting in the loss of contact.

Narrative

Aircraft X was a ZZZ [Airport] arrival that first came into our airspace through the XXXXX Sector Airspace. The aircraft was vectored for the ILS X at ZZZ [Airport]; and while on vectors; was descended down to 2000 feet on a SW bound heading. Training for the entire airspace was in effect; where both the trainee and trainer did not switch Aircraft X on XXX.XX; the ZZZ [Airport] airspace frequency. The transmitter for YYY.YY is located at ZZZ [TRACON]; and XXX.XX is located as WST (Westerly). With the low altitude of Aircraft X and distance from ZZZ [TRACON]; Aircraft X could no longer hear us. The trainee and trainer realized this when the aircraft did not respond to the vector or turn to base. This left the aircraft on a SW Bound heading towards ZZZ's [TRACON] airspace; where the MVA rises to 3000 feet for our approach control facility. Aircraft X did fly a few miles into ZZZ's [TRACON] airspace; with the 3000 feet MVA. A NORDO notification and point-out was accomplished with ZZZ [TRACON]. Upon being told of the situation; I immediately called ZZZ Tower; and asked them to dial in YYY.YY to turn the aircraft to a 020 heading; and switch the aircraft to XXX.XX. They were able to communicate this to Aircraft X to accomplish this. After the event and communications were established with Aircraft X; we reached out to ZZZ [TRACON] and found that the MVA in that area was 1600 feet for them. Although our MVA is higher for us; I do believe the aircraft was in no danger of flying into obstructions after finding this out. Suggestion: Continue training important habits and reminders to trainees on when to switch aircraft between frequencies.