As the update/installation progresses,  pop-up notifications will inform you if any extensions need to be Allowed via Security & Privacy. For detailed steps see the Troubleshooting section.
There are 2 extensions involved: SophosWebNetworkExtension and SophosScan. 
You can need to check Security and Privacy to ensure there is nothing pending to be allowed:

Apple menu  > System Preferences, click Security & Privacy, then select General
You are looking for this: (click on the images to see them in full size)

     

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. 

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.
Technical details can be found below: https://community.sophos.com/intercept-x-endpoint/big-sur-eap/f/recommended-reads/124249/big-sur-eap-known-issues-list 

 

We have identified a possible root cause for this issue. Expected fix to come out on 2021. 

 

After Installing/Updating to Sophos Home on Big Sur, you will see 3 Items in the new Launchpad/Applications folder:

• Sophos Home
• Sophos Scan
• SophosWebNetworkExtension

Big Sur will show all the apps in the new launchpad. There is no action needed, nor will these apps launch the antivirus. To launch Sophos Home, please see Locating Sophos Home in your system 
If desired, the apps can be moved to a folder to keep them all under one place:  https://support.apple.com/en-in/guide/mac-help/mh35840/mac