John Sergeant Nottingham's children in 1841 census are listed with surname "Frageon" - what does that mean? They were christened "Nottingham" and their mother is Dinah Nottingham. These are the correct children for the family in other respects.
Citing this Record
"England and Wales Census, 1841," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MQ1R-RFY : 13 December 2017), John Nottingham, Barrow, Lincolnshire, England; from "1841 England, Scotland and Wales census," database and images, findmypast (http://www.findmypast.com : n.d.); citing PRO HO 107, The National Archives, Kew, Surrey.
Aha! Elisabeth's married name is Farguson (from 1840) so maybe the census taker noted her name and mistakenly applied it to her younger siblings.
"England Marriages, 1538–1973 ," database, FamilySearch (https://familysearch.org/ark:/61903/1:1:NXQ1-T7Q : 10 December 2014), John Nottingham in entry for Thos. Holmes Farguson and Elisabeth Nottingham, 31 Dec 1840; citing Barrow-Upon-Humber, Lincoln, England, reference 21, index based upon data collected by the Genealogical Society of Utah, Salt Lake City; FHL microfilm 1,450,437.