My Honda Accord (2011 EX-L V6 Sedan) front left signal lights works as a signal. However, when the car is on night mode only the right signal says lit permanently, while the left is dark. The Left turn signal light only signals (blinks), but doesn't stay on. Any idea why?

That or is the right faulty for always being on when in night mode?

 

Did you check the bulb?
And night mode is when you turn on the running lights.