Using unattended-upgrades on Ubuntu  16.04 (and previous versions) set up to install updates on shutdown, flashplugin-installer always failed when downloading the new version. The reason probably is that on shutdown, the Internet connection is not available anymore. Up to now, I always updated flashplugin-installer manually. But today, I found a nice and simple solution:

Just get rid of the installer! In the Cononical partner repository, there is a package that directly contains the flash plugin. First, make sure you include the partner repository in your /etc/apt/sources.list, e.g. for xenial:

deb http://archive.canonical.com/ubuntu xenial partner
deb-src http://archive.canonical.com/ubuntu xenial partner

Then, update, get rid of the flashplugin-installer, and install the adobe-flashplugin instead:

sudo apt-get update
sudo apt-get remove flashplugin-installer
sudo apt-get install adobe-flashplugin

Hope this helps someone who also has problems with unattended upgrades and falshplugin-installer.