Unexpected file permission issue

After version 7.1.3, every attempt at updating to the latest version is met with this message, despite my having made no changes to the file system:

Update Failed: The update cannot be installed because we will be unable to copy some files. This is usually due to inconsistent file permissions.

The site has been hosted by Crazy Domains for a number of years.

The page I need help with: [log in to see the link]