Frances Jane Coolidge and Henrietta Louise (Avery) Coolidge were sisters, daughters of Henry Joseph and Lucy (Jones) Coolidge. Frances was born 14 Oct. 1791 in New Haven, Conn., and died 24 Jan. 1874. Henrietta was born 11 July 1802 in Savannah, Ga., and died 30 Mar. 1883 in Norwich, Conn. She married 6 Aug. 1832 in Norwich Charles Forsyth Avery of New Britain, Conn.
The sisters' parents are said to have moved to Georgia in about 1800. Henry died 7 Sept. 1803 in Savannah, Ga., after a short illness, aged 36 years. One newspaper account of his death states that he was "a native of New-London, Connecticut", another says a native of New Haven. I have not been able to establish his parentage. His wife, Lucy, was the daughter of Ebenezer and Elizabeth (Rogers) Jones, born 4 Apr. 1766 in Norwich.
Henrietta L. Avery and her widowed daughters Mary E. (Avery) Stedman and Harriet C. (Avery) Robinson were all living with Frances in 1870. I'm not sure if it was the same residence mentioned in your query, but "Old Houses of the Antient Town of Norwich," published in 1895, stated that Harriet was then the owner of her great-grandfather Ebenezer Jones' house: