However, Jack's leaving the fire causes him trouble later on. How does Ralph "assert his chieftainship" after the argument with the hunters? ... He gives the hunters the task of keeping the fire going. This gesture was important becasue it showed that Ralph was not afraid to step up and give orders.

How does Ralph assert his chieftainship after Jack apologizes?

The first thing Ralph does upon becoming chief is to grant shared power to Jack. Ralph is perceptive enough to notice Jack's mortification at being voted down, so he is "eager to offer something." He concedes that the choir still belongs to Jack and lets Jack choose what they are to be.

What does Ralph do at the end of Chapter 4 that asserted his chieftainship?

This is a reference to a specific moment in Chapter 4, where Ralph is said to have inadvertently "asserted his chieftainship" better through casual passive aggression than if he had thought about an alternative plan for days.