The 23-year-old was made Chelsea captain with Reece James out injured and Enzo Fernandez suspended by the club.
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Palmer didn’t have his best outing as Chelsea progressed to the FA Cup semi-final for the seventh time in the past ten seasons.
Speaking to reporters after the game, Rosenior insisted it was a “natural step” for Palmer to be handed the arm band.
“It's a natural step for Cole and where he is in his career. Cole is so brave, he'll take the ball on the pitch and if he makes a mistake, he'll take it again. That's what I want,” he said.
“You lead in different ways. Cole is a player in big moments who can win you games. Cole and Joao today, they're tracking back and defending. Cole led by example.”