Tower Controller reported an aircraft on final approach in IMC deviated off course and descended below terrain minimums. Radio transmissions from the aircraft were intermittent and broken.

Narrative

Weather conditions at the airport were dark - sun had set - VIS 7 SM; ceiling OVC006. I was all combined up at the time with all the positions in the Tower cab. Traffic was extremely light due to the weather; which had come in quickly over the course of about 15 minutes during the previous hour. I had noticed Aircraft X target flying VFR around for several minutes prior the incident. I recognized the call sign as one of our local planes; and was waiting for them to call up or for them to tag up with ZZZ1. Eventually; they tagged up and appeared to begin tracking the localizer approach inbound to ZZZ. I accepted the handoff while they were around 8 miles out; but they did not call me for another 2 to 3 miles at which point they were showing an altitude of about 019. Once they called in; the transmission was broken; with much noise. Nonetheless; I made out their call sign and cleared them to land on Runway XXL; also including the last PIREP I had received from Aircraft Y on the approach 15 minutes prior - reporting tops at 015 and bases at 005.I got no response from Aircraft X; so I re-keyed all my frequencies and tried again with the same transmission. If they responded it came in unreadable; so I was about to try again; when I noticed that they had begun a right turn off the approach path. I immediately told them to stop their descent due to the proximity of the ridgeline in that direction; and to contact ZZZ1 on a given frequency for better terrain advisories and/or resequencing and continued to repeat this for urgency. I called ZZZ1 over the shout line and told them Aircraft X would be returning to them; since I no longer knew what they were doing not following the approach; and also hoping that their transmissions would come in better with ZZZ1; their last frequency out there. The low altitude alert came on for Aircraft X; and I continued to tell them to stop descending. At some point; a broken stop descent... ZZZ1..." came through from them on the radio; but I continued to relay "stop descent" because their transmitted altitude did not appear to be rising yet.ZZZ1 also called over to have me climb them to 034; which I then continued to relay as well. Soon after; I saw their altitude slowly begin climbing and Aircraft X began a path away from ZZZ; plus ZZZ1 took the tag; so I understood that the aircraft was now with ZZZ1. I am reporting this because Aircraft X possibly came into close proximity with the terrain while inbound; with the lowest altitude I saw them at indicating 012. I wonder if perhaps; being one of our training aircraft call signs; the pilot was a student unfamiliar with IFR flight and did not understand IFR procedures. It is possible that; upon reaching the clouds on the approach; they began a turn off course to avoid; if that was what they were familiar with doing in past VFR flights. I'm not sure what I could have done differently in this scenario. Perhaps I could have thrown in a safe altitude to climb to; were I more familiar with the MVA in the immediate area."