As the update/installation progresses,  pop-up notifications will inform you if any extensions need to be Allowed via Security & Privacy. 
There are 2 extensions involved: Sophos Network Extension and SophosScanD. 
Check Security and Privacy to allow them as needed

(This applies to new users who are installing for the first time.
If you are re-installing, and there is nothing to allow, move on to the next step).

A message will appear on the top right, informing users to enable notifications. Click on the message and use the slider to turn them on. You may need to perform this twice

A Welcome to Sophos window will instruct the user to drag-drop extensions into Full Disk Access. 

(click on any image to enlarge it in a new tab) 

1 - Click on "Let's Start" on the popup message
2 - Follow the instructions to open Security & Privacy (by clicking on "Click Here")

3 - At the "Privacy" section, Scroll down to "Full Disk Access", click on the padlock at the bottom left and enter your Mac's username/password to allow changes, then drag and drop the "Drag this" image into the Security & Privacy window. 

4 - Once you have dragged the Drag This image into the Security & Privacy. Several Sophos components will be added to the Full Disk access. Click Done and Close Security & Privacy
Note: If the steps are not completed the first time, the pop up will show up on reboot, and also whenever a scan has to take place. 

Should the popup continue to appear after following the steps, please perform the troubleshooting listed here Troubleshooting Welcome to Sophos popup 

Once finished following all the steps, restart your Mac to complete the installation process.

The computer needs to be restarted to complete the installation. If the system is still vulnerable after a reboot follow the Troubleshooting section.

Network filtering applications such as Little Snitch or VPN software running in tandem with Sophos Home or other software with network filtering capabilities, may trigger errors or not work at all. If you experience issues with Sophos Home or another network based program, we recommend either disabling or removing the other program and re-testing functionality.