1. Investigate whether the Mozilla Maintenance Service is installed,
and also whether it is enabled.
* Control Panel >> Programs and Features
* Control Panel >> Administrative Tools >> Services
If the Mozilla Maintenance Service is installed and enabled, then
automatic updates should not need to ask for admin privileges.
The Firefox full installer installs both Firefox and the Mozilla
Maintenance Service. Someone could have unchecked the installer
option to install the Mozilla Maintenance Service. Someone could
have uninstalled the Mozilla Maintenance Service. And someone
could have disabled the Mozilla Maintenance Service, leaving it
installed but disabled.
2. If an automatic update failed and you then did as full install,
you may want to remove the downloaded automatic update files.
See these instructions in the mozillaZine knowledgebase:
Software Update not working properly
http://kb.mozillazine.org/Software_Update#Software_Update_not_working_properly
Follow the instructions to delete the "updates" folder and the two
files "active-update.xml" and "updates.xml".
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Kind regards
Ralph