Install Nginx in a cPanel server

1) Go to WHM > Tweak Settings and change the Apache port:

The port on which Apache listens for HTTP connections. Specifying a specific IP will prevent Apache from listening on all other IPs. (default: 0.0.0.0:80)

To:
0.0.0.0:8081

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2) Grab a copy of the nginx installation script:

mkdir /root/cpacct && cd /root/cpacct
wget underhost.com/nginxsetup.tar
tar xf nginxsetup.tar
cd publicnginx
./nginxinstaller install

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3) If you receive this error when trying to install:

access key doesn't exist create it in WHM

Then go to WHM > Cluster/Remote Access > Setup Remote Access Key and click on the "Generate New Key" button.

4)  At that point, then re-run "./nginxinstaller install" command. Rebuild Apache and then restart nginx and Apache
/scripts/rebuildhttpdconf
/etc/init.d/nginx restart
/etc/init.d/httpd restart