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Delete the folder administrator/components/com_admintools/sql, then retry the update.

This has nothing to do with our code, actually. It's a problem between your host's operating system and PHP, affecting Joomla's own code which is installing the updates. This happens when the ownership (NOT permissions) of the administrator/components/com_admintools/sql folder and / or its contents prevent the user PHP is running under from deleting or replacing that folder and its contained files. The deletion / replacements is something that Joomla itself does when installing an update and cannot be controlled by the code we publish.

FWIW it looks like Joomla started reporting this problem only after 3.9.5. Before that it would silently fail to replace the files, causing all sorts of weird issues which made us give the standard advice "try to install the update twice in a row, without deleting anything in between". That advice would work around Joomla not being able to copy files and execute the database table update despite the fact that the files were not ever copied to the destination directory, taking advantage of a quirk in how Joomla's extensions update code worked. At least now you get an error message that helps you solve the real root cause, even if it's a bit confusing as to where that message comes from.

You're welcome!