Magento – Magento Installer – File Permission Check Fail

I had been struggling with Magento-CE-2_1_3 installation without success.

The installation process is always terminated because of the File Permission Check

"File Permission Check

4 file permission not met. Hide detail

The best way to resolve this is to allow write permissions for files in the following Magento directories and subdirectories. The exact fix depends on your server, your host, and other system variables.
For help, see our File Permission Help or call your hosting provider.

"/var/www/html/mage2/app/etc" - Not writable, change the permissions. Show details
"/var/www/html/mage2/var" - Not writable, change the permissions. Show details
"/var/www/html/mage2/pub/media" - Not writable, change the permissions. Show details
"/var/www/html/mage2/pub/static" - Not writable, change the permissions. Show details"

Currently I ended up to change permissions to full access to all Magento content (files & directories).
e.g drwxrwsrwx. 2 apache apache 84 Jan 25 18:24 etc

Even if the content has full right the Installer is giving notice that there is no permissions to certain content.

Unfortunately I haven't found the solutions from Magento DevDoc or other sites.

All advices are welcome!

—added 30.1–

— added 2.2 —

After CLI installation the error log is filled with following when index page is opened.

[Thu Feb 02 12:32:24.441225 2017] [:error] [pid 6955] [client
127.0.0.1:56438] PHP Fatal error: Uncaught Zend_Cache_Exception: cache_dir "/var/www/html/mage2/var/page_cache" is not writable in
/var/www/html/mage2/vendor/magento/zendframework1/library/Zend/Cache.php:209\nStack
trace:\n#0
/var/www/html/mage2/vendor/magento/zendframework1/library/Zend/Cache/Backend/File.php(180):
Zend_Cache::throwException('cache_dir "/var…')\n#1
/var/www/html/mage2/vendor/colinmollenhour/cache-backend-file/File.php(87):
Zend_Cache_Backend_File->setCacheDir('/var/www/html/m…')\n#2
/var/www/html/mage2/vendor/magento/zendframework1/library/Zend/Cache.php(153):
Cm_Cache_Backend_File->__construct(Array)\n#3
/var/www/html/mage2/vendor/magento/zendframework1/library/Zend/Cache.php(94):
Zend_Cache::_makeBackend('Cm_Cache_Backen…', Array, true, true)\n#4
/var/www/html/mage2/vendor/magento/framework/App/Cache/Frontend/Factory.php(155):
Zend_Cache::factory('Magento\\Framewo…', 'Cm_Cache_Backen…',
Array, Array, true, true, true)\n#5
/var/www/html/mage2/vendor/magento/framework/App/Cache/Frontend/Pool.php(67):
Magento\Framework\App in
/var/www/html/mage2/vendor/magento/zendframework1/library/Zend/Cache.php
on line 209