Pinging a domain name uses ICMP. It does not open a TCP connection. See https://superuser.com/questions/674605/what-is-type-of-icmp-packets-tcp-or-udp for a quick explanation. Since your firewall, most likely an iptables-based one, is filtering TCP and UDP traffic (but not ICMP) you can ping but you cannot make HTTP requests (which use TCP traffic) to remote servers.
You should try using cURL from the command line to see if you can connect to a remote server over HTTP/HTTPS. For example, you can run
curl "https://cdn.akeebabackup.com/updates/backupwppro.ini"
to see if you can access our CDN. In order to receive and install updates you need to be able to access both cdn.akeebabackup.com and www.akeebabackup.com over port 443 (HTTPS). Please note that the IP address for the CDN is not guaranteed since it's a CDN and by definition it resolves to whichever server is closest to you (your server, not you the person).
Moreover, post-processing requires opening a connection to different domains. Google Drive requires access to both www.googleapis.com and www.akeebabackup.com over port 443. Also note that Google Drive MAY return a different URL it requires access to over port 443.
All these domains need to be accessible from your server, i.e. you need to allow outbound TCP connections to them. How you should do that is very well outside the scope of our support. It has everything to do with how the server and its firewall is set up. You need to ask your systems administrator to help you open outbound connections in your firewall to these domains.
Nicholas K. Dionysopoulos
Lead Developer and Director
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