Birth registrations:
Gertrude Coleman (no mmn listed), Dec qtr 1881 Spalding
Edith M__ Colman, c1887 Lambeth (poss died as Edith Colman aged 23, Dec qtr 1910 Spalding)
Ethel Maud Colman mmn Colman, Jun qtr 1890 Spalding
Lois Margaret Colman mmn Colman, Jun qtr 1894 Spalding
Thomas Henry Colman mmn Colman, Jun qtr 1896 Spalding
Joseph Frederick Colman mmn Coleman, Sep qtr 1899 Spalding
Beatrice Emily Sarah Colman mmn Coleman, Mar qtr 1903 Spalding
Frances Miriam Colman mmn Coleman, Sep qtr 1904 Spalding
Have you thought about obtaining the first two birth certificates to see if it reveals anything
that might be helpful regarding parent/s?
Have you found a baptism/christening record for Gertrude or Edith ?
Their sister
Beatrice Emily Sarah Colman (index only)
15 Feb 1903 Baptism Gosberton, Lincoln, England
Father:
Thomas Henry Colman Mother:
Mary ElizabethChristening RegistersWhat if the parents were unmarried? Would the father's name be listed in the parish register? Likely not. The mother may give his name and the cleric might record "reputed father", but this was not required. Under Civil Registration, the father had to be present in order for his name to be recorded on a birth certificate
An illegitimate child legally took its mother's surname. Whether he or she changed it later or not was entirely up to him or her, or those acting on the child's behalf
The above from Lincolnshire Church and Parish Records
http://www.genuki.org.uk/big/eng/LIN/churchrecs#MarriageLicGertrude's m/c I would assume would name Thomas Henry as her father, so would be of no help
Her children from her marriage (1899 Spalding) to Arthur James Facey all have
mmn Colman/Coleman1901c
https://www.familysearch.org/ark:/61903/1:1:XSDH-XPB1911c
https://www.familysearch.org/ark:/61903/1:1:X7YQ-LLG1901c gives Gertrude’s yob 1879 and 1911 yob 1882 pob Gosberton, Lincolnshire
Gertrude Facey died 1943 Spalding age 62
Freesearch 1939 Register surname Facey Spalding RD
Arthur 1877, Gertrude 1881, Alfred H H A 1913, Frank 1917