Worn disconnector or weak disconnector spring (thus no engagement)
Assuming the rifle has the "regular" parts and not some drop in "unit" thingy for a trigger group, then...
The disconnector IS engaging, but its not HOLDING long enough otherwise the rifle wouldn't double.
If the disconnector were not engaging at all, then the hammer would follow the bolt carrier down, and this almost never results in firing. (I say almost, because I try to never say never when it comes to firearms
)
The hammer is being held as the bolt closes, then is falling without the trigger having been pulled, firing round #2 but then it stops, not firing #3, and running away. The doubling is erratic, not consistent or predictable.
THis means the disconnector is holding the hammer normally and releasing it normally the first shot, (and the third and following shots, until it doesn't and doubles again,) with the trigger sear "taking the hand off" correctly and holding the hammer until the trigger is pulled.
Now, the second shot, the one being fired "automatically" has to be from either the disconnector letting go too soon (with the shooter still holding the trigger to the rear) OR the hammer & trigger not holding when the disconnector releases the hammer.
This can only be caused by worn parts, broken parts, misaligned/misassembled parts or some foreign matter interfering with proper operation. (assuming the shooter is not pulling the trigger the second time without realizing it).
Parts not assembled properly (pins walked out, spring in backwards, etc) is a simple check, if you know what you're looking at. Foreign matter (dirt, crud, gummy oil, etc) should also be visually obvious, but may not be, good cleaning is the fix.
Now, worn parts,...and in this case parts worn JUST ENOUGH to malfunction SOMETIMES but not ALL the time is a problem, and makes the rifle DANGEROUS.
If they're worn enough to malfunction some of the time, that means they are close to malfunctioning ALL THE TIME, and that is dangerous, it can mean a run away full auto gun, or a gun that fires with the safety ON and without the trigger being pulled. OR both together.
AND, there is no way to tell WHEN this could happen, it might happen the very next time you chamber a round. The gun should not be fired AT ALL, until this situation is corrected.
I have seen a trigger with the engagement nose broken, not completely all the way across but with about 3/4 of it broken off and gone. The tiny remaining edge would catch and hold the hammer, but part of the time the jar of the action closing would let the hammer "bounce off", fall, and fire the rifle. The same result can be from wear or breakage of the hammer or disconnector as well. There are numerous possible causes, but the most likely is something wrong with one (or more) of the 3 main parts involved.
