Skip the “Add/Remove programs”.

1_ Boot into Windows Safe Mode.
2_ Run the removal tool for AVG again.
3_ Reboot into Windows Safe Mode (do NOT skip the reboot).
4_ Run the removal tool for Avast, selecting the folder for Avast 6.
5_ Reboot into Windows Safe Mode (do NOT skip the reboot).
6_ Run the removal tool for Avast, selecting the folder for Avast 5 if the folder is still there. If the folder for Avast 5 is not there anymore, run the removal tool for Avast anyway (selecting Avast 5 but with no specific folder).
7_ Reboot into Windows Normal Mode.
8_ Delete any/all firewall rules related to security tools (Avast or any other security tool in the firewall rules).
9_ Install Avast anew, and reboot immediately the installation finishes.
10_ Let Avast connect and the firewall should ask you to rebuild the rules.
11_ Reboot once more time so to confirm it is working and report back for any possible inconvenient you had.