Further to my post I need to say that I have based my assumptions on the following:
Kate had a child Ernest in 1883 .
She was young so I presume got married when pregnant to Frank Hardy who lived near her in Langley Mill, he was a coal miner like her father William.
How she got together with Fred I do not know but it obvious from census returns that he moved about being a stone mason(as was my grandfather William) settling eventually in Darley Dale .
The problem is compounded by his change of name , being registered as Edward at death and in a census record. I have traced who I think were his parents to the 1861 census. They could have been Amos Scholfield and Sarah nee Smith. They moved from Todmorden to London see census records.
Why I think they are his parents is because Amos was a builder and Fred is more likely to have folowed this kind of trade
For info. for anyone looking
here are Fred & Kate on 1901/91 c
1901 RG13; Piece: 3266; Folio: 47; Page: 5
Fred SMITH 52 Todmorden
stone masonKate 36 Lanley Mills Notts
Ernest 18 Radford Notts stonemason
William 11 Yorks
Percy 6 Darley Dale
Bertha 4
Lily 3
William
BIRCUMSHAW lodger 66 Saleston Notts
1891 RG12; Piece: 2775; Folio: 105; Page: 5
Fred Schofield Smith 35 Todmorden stone mason
Kate 26
Ernest B 8
Edward A 4
William H 1
Kate widowed by 1911, dau Ellen Smith born 1907 Darley
and here is Amos & Sarah marriage & census detail
27 Sep 1847 St John the Baptist, Halifax Yorkshire, England
Amos SCHOLFIELD 22 no father named
Sarah SMITH 19 father John occ painter
Both single both res. Langfield
Witnesses John Holden (x) & John Thomas
1851 HO107; Piece: 2289; Folio: 229; Page: 29
Amos SCHOLFIELD 25 Walsden Lancashire Joiner & Builder
Sarah 22 Langfield Yorkshire
Sidney 2 ‘’
Sarah Ann 5 mths ‘’
1861 RG 9; Piece: 3234; Folio: 92; Page: 7
Amos Scholfield 35 Walsden Lancashire Joiner & Builder
Sarah 35 Todmorden Yorkshire
James E 8 ‘’
FRED 5 ‘’Emily 5 ‘’
Mary Ann Winsley 20
1871c RG10; Piece: 636; Folio: 9; Page: 12
Bermondsey London
Amos Schofield 45 Rochdale builders foreman
Sarah 42
Sarah A 20 Halifax
James E 17 Halifax Clerk
Emily 15 Halifax
FRED 15 Halifax boy in office
John 4 Hackney Middlesex
1881c Bermondsey London
https://familysearch.org/ark:/61903/1:1:QK6Y-ZVHHFred is no longer at home
family search have these christening entries surname
SCHOLFIELD parents Amos & Sarah
Walsden Lancashire
Fred 1855
https://familysearch.org/ark:/61903/1:1:NF7J-GFZEmily 1855
https://familysearch.org/ark:/61903/1:1:J7JY-1PPJohn 1877 born 1867
https://familysearch.org/ark:/61903/1:1:NTHR-J6YBaptisms parents Amos SCHOLFIELD occ Builder/Joiner & Sarah
Stone Cross Todmorden, Sidney 26 Aug 1848 born 3 Jul, abode Stack Hill in Langfield
Christ Church, Sarah Ann 3 Nov 1850, abode School Lane
‘’ ‘’ James Edward 7 May 1854 abode York Place
St Peter Walsden Emily &
FRED 25 Nov 1855 abode Todmorden in Langsfield
‘’ John 5 Aug 1877 born 21 Feb 1867