OK. I have a clue as to what happened from the Windows system event log. It says, "The avast! Administration Console service depends on the following service: MSSQL$SQLEXPRESS. This service might not be installed."
That is bad. I installed this using an existing SQL 2008R2 Express instance on another server, so that service is NOT installed on this server and therefore the avast! service cannot "depend" on it (that is, require it to be started prior to the avast! service starting).
Besides installing SQL 2008R2 Express on this server, is there any way to get around this install error??