This means you are missing a file from your site, namely administrator/components/com_akeebabackup/installers/brs.jpa. This file most definitely does exist in the packages we ship. Open our package ZIP file, then extract the com_akeebabackup.zip file you see in it. Look in the extracted directory backend/installers. As you see, the file is there. So, it's not an error on our part.
This means there are two possibilities here.
Either Joomla didn't copy this file, or something/someone else deleted it after installing the extension.
Try installing the Akeeba Backup package (pkg_akeebabackup-*.zip) twice in a row, without uninstalling it before or in between. This addresses the "Joomla didn't copy this file" problem.
If the problem happens again before you install the next update to our software, something/someone else deleted that file. The usual suspect is a host-provided "anti-malware" / "security" scanner which might not recognize the .jpa extension of the file and delete it. I've also seen some hosts which deliberately delete .jpa files because they do not want their clients to have backups they control (it gives clients negotiating power; your site is no longer at the mercy of the host).
Nicholas K. Dionysopoulos
Lead Developer and Director
🇬🇷Greek: native 🇬🇧English: excellent 🇫🇷French: basic • 🕐 My time zone is Europe / Athens
Please keep in mind my timezone and cultural differences when reading my replies. Thank you!