This is not a clean installation, or at the very least Joomla does not think it is.
When you install an extension, Joomla needs to decide if this is a clean installation or an update and apply a different set of database queries depending on this decision. It makes this decision by looking up information in two tables.
First, it looks in the #__extensions table, in this case: type=component, element=com_akeeba. If there is no record there it is a clean installation.
Note: #__ is meant to be replaced with your table naming prefix, the common few characters and an underscore all of your Joomla and third party extension tables start with.
If there was a record in the #__extensions table, it reads its extension_id. Then, it looks up the #__schemas table for a record with the same extension_id. If none is found it is a clean installation.
If a #__schemas entry is found, it reads the version_id, then it applies all upgrade database queries from the next version onwards.
For some reason, Joomla believes that you have Akeeba Backup already installed, i.e. there is an #__extensions table and a #__schemas table record for it in the database. As a result it tries to apply the update database queries which (reasonably) assume the database tables for Akeeba Backup exist. Since they do not exist, Joomla's extensions installer errors out.
The solution in this case is to remove the #__schemas and #__extensions table records for Akeeba Backup, then retry the installation.
If System, Manage, Extensions shows Akeeba Backup as installed: uninstall it. This will do the trick. Note that you may see it listed as COM_AKEEBABACKUP if the language files for Akeeba Backup are missing.
If that page does not show Akeeba Backup as installed or you cannot uninstall the component we will need to do it The Hard Way: editing the database directly. If it comes to that, let me know and I will ask you for connection information to your site.
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
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