DickW is on the money.
Fuses are irrelevant as is current from a 12 volt battery (it could deliver over 500A for the starter motor at initial switching or thousands of amps if short circuited across the terminals).
Seems to me that you short circuited the contact breaker to ground directly from the battery! You perhaps need to understand Ohm's Law before doing thigs like that, if that was what you did.
The coil is an inductive load, but while final current could be 5A, it would have a rising current, the rate of which depends on the time constant of the coil. Average running current would be far less as the dwell angle will also be taken into account.