Here is an article about extraversion and leadership ("Stop Stealing the Spotlight: The perils of extraverted leadership").
According to the article the "real core" of extraversion is the tendency to enjoy and attract social attention. They take initiative, bring the vision, assertiveness, energy, and networks necessary to give employees direction.
But what happens when an extravert comes in a leadership position and he must lead modern pro-active employees? They will more often than introverted leaders ignore their suggestions or take credit for their ideas.
So the article - that is based on some scientific experiments - recommends extraverted leaders for passive employees and more introverted leaders for pro-active employees.