Normal safe mode doesn't have networking and that includes internet connection. If you need internet while in safe mode, you have to boot into, "Safe mode with Networking" (directions in same link as before.) However, if you add internet connection when your antivirus isn't running, that doesn't seem very safe to me so I'd avoid that unless necessary somehow.
I did some googling and found posts on the Avast forums that said that some versions of Avast won't run in safe mode. Instead, you can run a boot-time scan (which they claimed was even better.)
https://www.avast.com/no-no/fa...
I don't have Windows 10 or Avast so I can't test anything out for you. I would if I could.
Also, I'm not an expert with computers so my knowledge is limited. I wish that I knew more to help out.
