I have not found Karen moving to or from Hurum in Buskerud county, but this was not always noted.
When Caroline was baptized in 1876 Karen was married to a man with surname Amundsen.
This must be them in the 1865 census:http://digitalarkivet.uib.no/cgi-win/WebCens.exe?slag=visbas...
Johan was a murer = bricklayer and they live in Lakkegaden. Ole might have been a friend and colleague.
Here Karen is living with to daughters in the 1875 census. The daughters surname is formed from the first name of Karen's husband (or ex-husband. I think the f means she is divorced):http://digitalarkivet.uib.no/cgi-win/WebCens.exe?slag=visbas...
They live in Maridalsveien 86.
There are at least 6 possible Ole Nielsen/Nilsen in the 1875 census for Oslo only; so we need more information to know for sure who is Caroline's father.
Do you know when and where she was confirmated?