You are right. When you shut it down, the Avast icon disappears and you no longer have access to to the Avast interface.

So what does this mean? Does it mean that the whole Avast program is not stored on the home computer, and you have to allow this connection to Google in order to see the Avast interface?
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What does Google have to do with Avast? Can you elaborate please. Windows allows you to hide or show icons. Been like that since the Win 98 days. So let your OS hide the icon. Its that easy.