Use only good quality plug cos overheating is very common problem.
Fiat 500e Level 2 Battery Charger problems
Most common Fiat 500e Level 2 Charger by FCA problem is neutral and phase wire swap. In this case, the charger will not pass the initialization test. The effect is visible on the pictures below
Another common problem in old European installations is the lack of protective wire. This can be bypassed by combining a protective wire with a neutral one.
To diagnose all problems with charging cable You should use proper OBDII Fiat 500e diagnostic tool.