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Joomla! version: (2.2.4)
PHP version: (unknown)
MySQL version: (unknown)
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Admin Tools version: 2.2.3 upgrading to 2.2.5
Just to let others know I found the way to prevent the message that I was getting that said "Unable to open archive Extracting the update package failed. Please try updating the extension manually".
It's not enough to just enter your ID via the Options area, you also have to get the Live Update feature to use it. On closer reading of the instructions where we download our product ID number (https://www.akeebabackup.com/instructions/1436-admin-tools-download-id.html), the second to last paragraph was a little confusing to me in that it said click a button next to "Component Parameters" which I couldn't find, but basically when you get to the Live Update Admin Tools page (via the button under "Updates"), the instructions say to use the button marked "Refresh update information". In my case the information in the "Direct download URL" appended my ID number and then the error went away and the update installed perfectly.
Hope that helps someone who, like me, didn't properly follow the full instructions.