To do something perceptively is do it with insight and intelligence. A good doctor will perceptively examine patients.
To perceive is to be aware, and someone perceptive is considered very aware, smart, and insightful. Therefore, to act perceptively is to show smarts and understanding. A quarterback perceptively changes the play if he sees a blitz coming. A President perceptively consoles the country after a tragedy. The word perceptively describes actions that show great understanding and sensitivity to people and situations.