DIAGNOSIS This is a classic symptom of a failing ignition coil. When the coil heats up, its internal winding fails and kills the spark. The engine shuts off abruptly. When it cools, contact is temporarily restored.
FIX Confirm the diagnosis with a spark test when the saw is hot and failed. Remove the spark plug, reconnect the boot, ground the plug against the cylinder, and pull the starter. Observe for a strong blue spark. If there is no spark when hot, replace the ignition coil. The correct part is Husqvarna 544018401.
IF THAT DIDN'T WORK The problem could be a faulty ignition switch or a break in the stop switch wire. Disconnect the two wires from the ignition coil that go to the stop switch. Try starting the saw. If it runs and does not shut off prematurely, the switch or its wiring is faulty and must be replaced.
HEADS UP When replacing the coil, the air gap between the coil legs and the flywheel magnets is critical. Use a business card or the specified 0.3mm feeler gauge to set the gap. Incorrect gap causes weak spark or no start.