You need to be able to contact www.akeeba.com and cdn.akeeba.com over HTTP (TCP 443). Here's the thing. I cannot give any IPs for them. The former is served by CloudFlare and the latter by Amazon CloudFront (and will probably be migrated to CloudFlare as well). The IP they resolve to might be different depending on where you are. Your systems administrator needs to use dig on the server to determine the IPs.
This is not all! If you plan on using any kind of remote storage service your server needs to be able to communicate with the respective storage engine. Since these are third party services we CAN NOT provide IPs or even domain names since neither is under our control. If your backups fail to transfer anything to remote storage you need to have your systems administrator check what he's blocking and unblock it. We will NOT be able to provide any support beyond telling you "your server is blocking outgoing communications".
Generally speaking, what your systems administrator is trying to do is going to break your site six ways to Sunday. This kind of blocking only makes sense if there is absolute control over everything that runs on the site, with a very clear knowledge of every hostname and IP the software on your site needs to connect to. Trying to do that with a generic CMS and mass distributed software is... how do I put it mildly... misguided? Suicidal? Something like that.
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!