I have found the following family information for the only Fern Brown born in Massachusetts in the correct time frame. The births and second marriage of Annie RAE are records obtainable from the Archives. The birth record would give the father's name. I have also included the available census data and put it in a timetable format and included the contact information for the archives.

Massachusetts Archives
150 Mt. Vernon St.
Dorchester, MA 02125
(617) 740-2600
http://www.mass.gov/dph/rvrs

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Brown, Ruth A. b Lawrence 1894 Vol 439 page 425 Births
Brown, Fern b Lawrence 1897 Vol 466 Page 492 Births
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1900 census
Children's Home, Lawrence Essex MA
Brown, Ruth E b June 1894
Brown, Fern A b Dec 1898

Father born Vermont
Mother born England

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1900 census Lawrence Essex MA
Brown, Annie, lodger, b Nov 1872 England, mar. 10 yrs, two children,
NOTE: husband not in household

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1902 Vol 524 page 427 Marriage
Lawrence, Essex MA
Cockroft, William J.
Brown, Annie (maiden name RAE)

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1910 census
Abington, Plymouth, MA
Cockroft, William
Cockroft, Annie b. abt 1875 England, Imm.1880
Brown, Ruth b abt 1895
Brown, Fern b abt 1898

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Jo-Ann