Try changing the access permissions of the /var/www/html folder so that other users can read and write there. Here is a screenshot: http://i.imgur.com/ydsyujW.png

Thank you.
That worked, but when SET does that action it should also copy files to the /var/www/html folder and not the /var/www folder.
So I had to move the files there too and then it would work.
Is there anywhere in SET I can fix that? The website clone part?

Edit /etc/setoolkit/set.config and under "### Path to the Apache web root.", set APACHE_DIRECTORY=/var/www/ to APACHE_DIRECTORY=/var/www/html/ . This bug is also fixed in a newer version of SET https://github.com/trustedsec/social-engineer-toolkit/issues/149 .

I just pushed the new version 6.5.6-0kali1 of set in sana-proposed-updates.
You will have to add sana-proposed-updates in your /etc/apt/sources.list
Please test and let me know if it fixes this issue.

The problem with the cloned website location is fixed, but I still have to manually change the /var/www/html folder permissions so that other users than root can write and delete files there. Also when the a website is cloned, the automatically created harvester.txt remains in /var/www/, but this is not a big problem, because I can delete it manually.