This seems to fit for Catherine
1861 ST. MARTIN, JERSEY, CHANNEL ISLANDS
BURNEY
Catherine, age 31, naval officers wife, born England
Cecil, son age 2, born St. Martin, Jersey
Kate, daughter, age 6, born St. Martin, Jersey
Ernest H., son, age 8 months, born St. Martin, Jersey
PUCKETT
Annie, age 28, widow, sister, born England
1871 GREENWICH, KENT, ENGLAND
BURNEY
Charles, age 45, staff commander on the...............list, born Devonport, Devon(could not read the word before list)
Catherine, age 43, wife, born Middlesex, London
Catherine E. age 17, daughter, born Gorey, Jersey
Cecil, age 12, son, born Gorey, Jersey (indexed as Ceal)
Ernest H., age 10, son born Gorey, Jersey (indexed as Emily)
Percy H. J. age 8, son born Gorey, Jersey (indexed as Perey)
Annie L., age 5, daughter, born Gorey, Jersey
BUCKETT
Annie, age 38, sister in law, widow, born Middlesex, London
1881 GREENWICH EAST, GREENWICH, LONDON, ENGLAND
GREENWICH HOSPITAL SCHOOL
BURNEY
Charles, age 53, superintendant, commander Royal Navy (retired is crossed out) born Devonport, Devon
Catherine, age 52, wife, born London
Catherine, age 26, daughter, born Gorey, Jersey
Percy, ge 18, son, liet. Royal artillery, born Gorey, Jersey
England BMD has death for a Charles Burney born about 1826 died 1887 Greenwich, London
and England calendar of wills has
The will with a codicil of Charles Burney , late of Greenwich Hospital School, Greenwich in the County of Kent, retired Captain Royal Navy C.B. Superintendent of the said school who died June 19, 1887 at Greenwich Hospital school was proved at the Principal Registry by Cecil Burney of Greenwich Hospital school, a Lieutenant Royal Navy, the Son, and Albert Escott of Greenwich Hospital School, Schoolmaster the executors.
Personal estate 4135 pounds 16 shillings 3 d
Resworn June 1888 3,653 pounds, 10 shillings 10d
there is also at England BMD a death for a Catherine Burney born about 1828 died 1883 Greenwich, London
and Calendar of wills
June 23, administration *with the will) of the Personal estate of Catherine Burney (wife of Charles Burney) formerly of Gorey in the Island of Jersey but late of Queen's House Greenwich in the County of Kent who died May 11, 1883 at Queen's House and was granted at the Principal Registry under certain specialties to the said Charles Burney of Queen's House Staff Commander in Her Majesty's navy the sole executor.
I looked for marriages for the Burney children: Catherine, Cecil, Ernest, Percy and Annie
I found your grandmother Catherine and see she married John George Augustus Albert TREWIN
March 17, 1883 St. Alphege, Greenwich
he is shown as a mariner - bachelor, of full age, father John Geak Trewin, late of Royal Navy
she is shown as spinster of full age , father Charles Burney, Commander, Royal Navy, C.B.
both Catherine and husband shown as residence at time of marriage as being Royal Hospital School
witnesses Annie Burney and other name hard to read but looks something like Francis G. Hay ??
** I thought the late of Royal Navy meant John was deceased but it does not -see his death further down
I did not find any of the others in London marriages
TREWIN
You asked about Augusta Georgina - there is a Wills calendar entry
TREWIN
Augusta Georgina, otherwise Augusta of 23 Warwick - place Leamington Spa, spinster
died 18 July 1953 at Warneford Hospital, Leamington Probate London 9 September to Henry Fergus Graham, solicitor Effects 460 pounds 14 shillings 11 d
this looks like her brother and parents
1861 TRANMERE, BIRKENHEAD, CHESHIRE, ENGLAND
TREWIN
John, age 35, master mariner, born Outhorn, Yorkshire
Georgina, age 30, wife, born Middlesex, London
John George, age 4, son born Gorey, Jersey, Channel Islands
1871 ST. SAVIOUR, JERSEY, CHANNEL ISLANDS
TREVIN (as indexed)but TREWIN in the image
John Grake, age 62, master mariner born England (in the image age is 42)
Georgina, age 40, wife, born England
John G., age 14, son born St. Saviour, Jersey
Harry R. age 10, son, born Cheshire, England
Augusta G., age 3, daughter, born Grouville, Jersey
Not sure what happened to Augusta after 1871 and before her death but cannot yet find her in 1881 when she was about 13. This looks like her father
1881 ST. LEONARD, BROMLEY, LONDON
TREWIN
J. age 52, widower, retired sea captain born Hawthorne, Yorkshire
there is an initial after J. but hard to read - indexed as J.T.
and calendar of wills - March 7, 1889 administration of the person estate of John Geake Trewin, late of 305 East India Dock Road in the County of Middlesex, master mariner, a widower who died 12 Feb 1889 at 305 East India Docke Road was granted at the Principal Registry to Augusta Georgina Trewin of Eden Villa St. Brelade's Bay in the Island of Jersey, spinster, the daughter and one of the next of kin
personal estate 179 pounds 1 shilling 7d
AND
Calendar of wills - TREWIN Georgina Mary Ann - 15 August, administration of the effects of Georgina Mary Ann Trewin (wife of John Geake Trewin) late of St. Saviour's in the Island of Jersey who died 9 June 1877 at St. Saviour's was granted at the Principal Registry to the said John Geake Trewin of Queen's House Greenwich in the County of Kent an officer in the Merchant;s Marine Service - effects un 100 pounds
OK here's Augusta in 1891
1891 ST. CLEMENT, JERSEY, CHANNEL ISLANDS
TREWIN
Augusta age 23, visitor, living on her own means, born Grouville, Jersey
she is living in the house of Mary Ann De Rue age 56
1911 - SARK, CHANNEL ISLANDS
LOMA
Fanny Conway, age 70, single, private means, born France, father born Lancashire
TREWIN
Augusta, age 37, visitor, single, niece, private means, born Grouville, Jersey, father born Cornwall
and this seems to fit for Augusta in 1901
1901 GROUVILLE, JERSEY, CHANNEL ISLAND
LOMAX
Fanny C. age 65, living on own means, born France (British subject)
TREWIN
Georgina, boarder, single, age 33, living on own means born Grouville, Jersey
England select births and christenings -John Geake Trewin - baptism Oct. 2, 1825 Owthorne, York, England - parents John and Sarah