How are emergencies handled during In-Seat Rest? Consider a Boeing 737 cruising over the Atlantic when an engine flames out. The procedure here is clear: the pilot flying should continue flying the plane, while the pilot not flying should pull out the QRH and follow the checklist for engine flameout. What happens, though, if the pilot not flying is engaging in Flight Crew In-Seat Rest? I assume they're woken up, and then I see a few options:


the pilot flying continues to fly while the pilot not flying wakes up and overcomes their sleep inertia. After 20 minutes, they go through the engine out checklist
the pilot flying continues to fly while the pilot not flying immediately goes through the engine out checklist, despite the risk of sleep inertia
the pilot not flying immediately starts flying despite the risk of sleep inertia, and the pilot flying goes through the engine out checklist
the pilot flying flies the plane while going through the checklist as the pilot not flying wakes up


These all sound like awful options. So  what would actually happen in such a situation?

Related: Is a pilot allowed to sleep during a flight?
Option 2:


  the pilot flying continues to fly while the pilot not flying
  immediately goes through the engine out checklist, despite the risk of
  sleep inertia.


When waking up to an emergency situation your body will start to produce adrenaline which gets you over the sleep inertia very quickly. If the situation is not clear to the PNF, he will be briefed by the PF. 

Besides that, in a checklist flow, both pilots are involved and crosscheck each other's actions.