Please bear in mind that the extension update is performed by Joomla itself. The error message comes from Joomla itself as you can see. The problem has to do with the combination of the file's ownership and permissions and how it interacts with the user your web server runs under. This is an operating system level issue, not something we can programmatically change.
If you are not sure what I am talking about please read the Security chapter of Akeeba Backup's documentation where I explain in great detail how users, groups, ownership and permissions interact. Yes, I know you are trying to install Admin Tools and I told you to look at Akeeba Backup's documentation. The chapter I am asking you to read has nothing to do with Akeeba Backup; it's about your operating system.
There are two ways these issues can arise:
1. Your web server runs under a user (e.g. "www-data" or "nobody") but your files, having been transferred by FTP, are owned by a different user (e.g. "myuser"). If the third digit of the octal representation of permissions is not 7 your web server cannot write to the file. Write permissions are also required to delete the file.
2. The file is owned by the same user as the one your web server runs under but it has the wrong permissions, e.g. 0400 (read only). In this case Fix Permissions will work but it has to run before trying to write to, rename or delete the file in question.
Given that you already tried using Fix Permissions and it had no effect (it's incorrect to say it "didn't work" -- it worked, but was ineffective because it's not applicable to your use case) I assume that you have the first problem.
The simple way to temporarily address it is to change the ownership of all files and folders under your site's root to the same user your web server runs under. However, as soon as you try to use FTP / SFTP you will end up in ownership and permissions hell.
The correct way to address it is to use FastCGI on your web server so each site runs under the same user as the one owning its files AND have all files and folders of each site on your server owned by the same user you use to FTP / SFTP into it. Also, never ever use SFTP as root, and in no circumstances whatsoever even think about running the web server as root.
In any case, these are operating level / server maintenance problems, not something that our software has any effect on. Please ask your host or systems administrator to help you. We unfortunately cannot offer server setup help beyond pointing out what the problem is for obvious reasons.
Nicholas K. Dionysopoulos
Lead Developer and Director
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