But every time I go on line Avast attempts to install again even though I have verified that the program and the update are current.
Un-check the automatic updates.
Note, avast is fully configurable. It will test for a connection each 40 seconds and when succeed, try to update. Then wait 4 hours to start the process again. You can set the updates to manual or 'ask for'. See 'Settings' link on my signature or the avast settings.
Is there some .exe file that triggers a continuation of a discontinued download and attempts to reinstall the Avast software?
No John, it's the normal update schedule if you set for automatic.