WordPress
From Kb
For auto updating, the following permission changes usually work. Run these commands from within the WordPress root directory:
find . -type d -exec chmod 775 {} \; find . -type f -exec chmod 664 {} \;
You may also need to add this directive to the wp-config.php file:
define('FS_METHOD', 'direct');
The webserver user may also need to own or be the group for all of the WordPress files:
chgrp -R apache *
Mor information on WordPress permissions: Changing File Permissions
SSL Bypass
Due to SSL vulnerabilities a lot of sites will disable older versions of SSL (as they should). This can cause some problems when updating if software that uses SSL is not upgraded.
As a temporary bypass for the SSL problem, this can be used: http://apipeandakeyboard.com/2015/09/16/download-failed-error0d0890a1asn1/