Unable to update to WordPress 4.7.2 - Error: inconsistent file permissions

I had similar issues with updating on my local. I ended up using wp-cli: http://wp-cli.org/

If you have ssh access it is worth checking out. I ran a sudo wp core update --allow-root (not recommended for live site) and its updated with no permission error.

For me, the command below worked. In the example below, replace "www-data" with the name of user under which the web server service runs. Also replace "/path/to/site" with the actual path to the root of your WP site.

chown -R www-data:www-data /path/to/site

IMPORTANT Be sure to revert this afterwards by granting ownership back to a regular user (not the web service user) otherwise it will not be secure at all. Also, leave the web service user as owner of the wp-content/uploads folder so that users can upload media.

chown -R user:group /path/to/site
chown -R www-data:www-data /path/to/site/wp-content/uploads

Have you tried manually uploading the updated WordPress files, via a programme such as FTP (Filezilla)? Do you still get the same problem?

Update:

Run the following

Reset the permissions of all files to 664:

find /path/to/site/ -type f -exec chmod 664 {} \;

Reset permissions of directories to 775:

find /path/to/site/ -type d -exec chmod 775 {} \;

Reset the group to the wordpress group (or whatever group makes sense for you)

chgrp -R wordpress /path/to/site/