DB/etc backup file permissions

I have tried to backup the db (as well as other backups – e.g., wp-config) to not avail.

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DB Backup failed. Please check the permissions of the backup directory.
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I have tried changing the permissions to 777, 666, and back to the pre-set 755. None of the permission settings will allow the plugin to conduct the required backups.

Any recommendations on the next step – if any?

http://ww.wp.xz.cn/plugins/all-in-one-wp-security-and-firewall/