Cannot install Docker extension in Plesk: Sub-process /usr/bin/dpkg returned an error code (1)

Symptoms

1. Server is running on Virtuozzo 6 container. 
2. Cannot install Docker extension:

   Execution of /opt/psa/admin/plib/modules/docker/scripts/post-install.php failed with exit code 1 and the output: 
   ERR [extension/docker] Execution installer has failed with exit code 1, stdout: Setting up docker-engine (17.05.0~ce-0~debian-jessie) ... 
    Job for docker.service failed. See 'systemctl status docker.service' and 'journalctl -xn' for details. 
    invoke-rc.d: initscript docker, action "start" failed. 
    dpkg: error processing package docker-engine (--configure): 
    subprocess installed post-installation script returned error exit status 1 
    Errors were encountered while processing: 
    docker-engine 
    , stderr: E: Sub-process /usr/bin/dpkg returned an error code (1) 
    ERROR:root:Command '['apt-get', '-qq', '--assume-yes', '-o', 'Dpkg::Options::=--force-confdef', '-o', 'Dpkg::Options::=--force-confold', '-o', 'APT::Install-Recommends=no', '--purge', 'install', 'apt-transport-https', 'ca-certificates', 'dmsetup']' returned non-zero exit status 100 
    exit status 1

3. Error in /var/log/plesk/panel.log:

   Errors were encountered while processing:
    docker-engine
   , stderr: /dev/mapper/control: open failed: Operation not permitted
   Failure to communicate with kernel device-mapper driver.
   Check that device-mapper is available in the kernel.
   Incompatible libdevmapper <C8> and kernel driver
   Command failed

4. Cannot install Plesk extension via Extensions > My Extensions > Docker > Go to Extension:
   

   PLESK_ERROR: Caution: The local Docker node is missing. See this KB article to solve this problem.
   Remote nodes can only be used with the paid version of the extension. You can buy the extension here.

Cause

Virtuozzo container is not configured properly.
There are known issues with installing Docker inside the container.

Resolution

Contact hardware node administrator to configure the container or Virtuozzo Support Team.