Which computer has the Internet connection that the other shares ?
Perhaps there is something relating to updating through the network and wireless router, sorry I don't use a network so can't suggest what to check.
If as you say the two computers use the same key, and you talk of almost a year ago, if your registration is out of date (although the Home version should last for 14 months) this may have nothing to do with it as you say this has been the case for most of the time on one system.
However, it wouldn't hurt to re-register. Complete the
avast! 4 Home Registration form and you will receive an email from register @ avast.com (without the spaces) with your registration key.
If computer 2 does update automatically and the settings are essentially the same.
Try a repair of avast on the system that doesn't update automatically. Add Remove programs, select 'avast! Anti-Virus,' click the Change/Remove button and scroll down to Repair, click next and follow.
If that doesn't work try, uninstall, reboot, install, reboot.