The Karls I am related to come from Chotieschau in Bohemia which is now Chotesov in the Czech Republic. I have pretty reliable information on what they did in Chotesov and how they left but there is some folklore as well. I have heard that for our family name Karl was a made up name that one of our ancestors adopted when they fled Germany to avoid being conscripted to the German army many years before. I don't know whether this is true or not. The Karl that arrived in NZ originally in 1863 was Joseph Karl. His brother Wenzl arrived in 1879.
Joseph (born 1825) and Wenzl (born 1827) were the sons of Thomas Karl and Anna (nee Schramm). I don't know if they had other children. Thomas was the son of Christopher Karl of Auherzen (not far from Chotesov) and Margaretha (nee Schuster or Schusterinn) from Mantau (now called Mantov and also not far from Chotesov). Anna Schramm was the daughter of Andreas Schramm and Magdalena (nee Bauer or Bayerin) both from Wasser-Aujezd (now Vodni Ujezd also not far from Chotesov).
Joseph was married to Anna (nee Turnwald) the daughter of Anton Turnwald of Chotesov and Anna (nee Gruber) of Stich (this may be Stod near Chotesov but not sure). Joseph and Anna had ten children Jack, Charles, Vincent, Andrew, Edward, Mary, Annie, Francis, Joseph II and Emily. I am descended from Andrew.
The following link is from the Puhoi Historical Society http://www.puhoihistoricalsociety.org.nz/
where there is quite a lot of info about how the Karl's settled in NZ. In the History tab there is quite detailed info about the journey out to NZ and the route they took etc. The Karl's did not settle at Puhoi but left for Ohaupo which is further South in the North Island of NZ.
My mother somehow obtained all the birth and marriage info for the above Karl's before they left Bohemia which I have copies of. You can get the marriage info on LDS but I can't tell whether or not Wenzl and Joseph or any of the others in the family history had brothers or sisters.
Hopefully this is enough that you can see whether there is a connection. I can followup with my mother on where she got the records from if that's helpful.