Sounds very much like you have my worst nightmare, a wire that is broken at the crimp connector
When faced with this situation is use short jumper wires inserted into the switch plugs
And in a situation like yours I unplug every switch, jump them all then holding the plug push & pull each & every wire till I hear the solenoid engage .
IF there is no joy then I use a longer wire to bypass the mowers wiring so in this case 2 jumpers from the S terminal, touch the end of the long one to the trigger terminal on the solenoid
If the engine does not crank then check the ground wire / connection on the solenoid
Engine now cranks you probably have found the problem, but check all of the others as well by moving the long jumper to the PTO switch, & touching the other end to the solenoid trigger terminal
Engine cranks then go to the clutch brake switch and do the same test
Now to do this you have to download your parts book from Husqvarna and right up the front is the wiring diagram
The only connection you need to make on the key switch is B to S so the engine will not actually start because the fuel solenoid is not energised but the engine should crank
Now a word of warning , the 4 pin plugs carry battery voltage for cranking the engine and a ground circuit to kill the magneto .
If you accidentially connect power to the magneto circuit then you will blow the magneto coils in less than a microsecond
All of the wires you are jumping should be white
If I have the correct wiring diagram in front of me your mower has that 4 pin seat switch with the anti-tamper shorting strip that is not in the cranking circuit but is in the ignition circuit so the engine will not start unless there is some one in the seat
The above assumes that you mean the engine will not revolve with the starter motor as people have different understandings of the terms "crank" , "fire" , "start" & " run"
Let us know how you go
These problems are a PIA and can take forever to find
If it ends up being a bad crimped terminal then you will ahave to pull the terminal out of the plug, crimp a new one on and lock it back into the plug .
