Please try to remember that you have the mower in front of you and we have words on a screen.
So you are saying that if you bridge the terminals on the top of the solenoid & the mower cranks as it should with the key but the key does nothing ?
Now I can not find a wiring diagram for your mower but the genetic tests go like this.
If it has1 trigger wire at the bottom of the solenoid, jump the battery wire to that terminal.
If Solenoid is good then it will crank
If solenoid is good check that wire ( should be orange pink or yellow ) for battery voltage when the key is in the S position
The genetic system is key S to PTO ( 2 pin pair ) , through that to parking brake to both lap bars to the solenoid
I use short wires to jump each plug starting with the key switch plug B to S which remains then all of the others till the engine cranks.
If you leave B & S connected then you will hear the solenoid trip which saves you growing a 3rd paw to work the key
Now if it has 2 trigger wires, then when you turn the key to S one should go battery voltage & the other ground.
This system is basically the same except that usually the ground is switched vial the lap bars
So power goes key switch PTP switch brake switch solenoid
Ground goes ground lap bars ( both of thm ) solenoid
If still no joy then I use longer wires to go from one switch block to the next switch block which finds broken wires
Usually Husqvarna uses yellow, orange or pink for power to the solenoid and black green or brown for ground circuits
as a precaution when dealing with unknown wiring I remove the kill wires from the coils to prevent accidentialy applying power to it which kills the coils instantly .
