The firewall is Sygate 5.6 Build 2808. "Allowed" Avast programs are ashServ.exe, ashMaiSv.exe, ashWebSv.exe.
Into the firewall settings, the following programs should be allowed to connect:
C:\Program Files\Alwil Software\Avast4\ashWebSv.exe (avast! Web Scanner)
C:\Program Files\Alwil Software\Avast4\ashMaiSv.exe (avast! e-Mail Scanner Service)
C:\Program Files\Alwil Software\Avast4\Setup\avast.setup (avast! Update executable). This is a temporary file that just appears when an update (check) is about to launch, and disappears again afterwards.
Don't need rights to connect:
C:\Program Files\Alwil Software\Avast4\ashServ.exe (avast! antivirus service)
C:\Program Files\Alwil Software\Avast4\aswUpdSv.exe (avast! Update Service)
Sygate could be messing the connection... delete the avast entries into Sygate settings, browse and update to 'add' them again.