William Stanfield Abernathy (1834-1896) married Margaret A Fitzpatrick (1838-1907) on 29 Jun 1858 in Maury County, TN. Both are buried in the Old White section of the Maplewood Cemetery in Giles County TN, along with at least 3, possibly 4, of their sons. To the best of my knowledge, they had 8 sons including one who died, possibly un-named,in infancy, and no daughters.
My 3rd-hand notes cite records of Elizabeth Denty Abernathy to the effect that sons John Morgan (Jms M on the census record you quoted), James and Irvine died unmarried, while son Robert married Mary Wells, and died in Pulaski, leaving two sons, Edwin and Robin. Morgan, James, and the eldest son, Samuel who died before reaching his teens, are all buried in the Old White section of the Maplewood Cemetery in Giles County TN.
As you noted, Ernest & Lawrence (identified as Lamonica on cited census) went to Oklahoma. Lawrence was unmarried as of 1920 census, and Ernest died in 1915 and his only son was killed in an auto accident.
To the best of my knowledge there may not be any living descendents of William Stanfield Abernathy & Margaret A. Fitzpatrick.
William Stanfield Abernathy was one of 9 children born to William David Abernathy (1800-1849) and Mary Beckwith Hardaway (1808-1872). His lineage (and possibly hers) streaches back into Colonial Virginia pre-Revolutionary War times. He was born in Brunswick County VA and she in Dinwiddie County VA. Both are buried in the William D. Abernathy Cemetery in Giles County TN.
William David Abernathy was a son of Jesse Abernathy (1772-1807) & wf Mary/Martha Wall, a grandson of Liles & Elizabeth Abernathy of Dinwiddie County VA, and gr-grandson of David Abernathy (d. 1784) & Ann Liles of Dinwiddie County, beyond which there are no paper records extant to verify the lineage. However, Y-DNA evidence has confirmed this lineage is very closely related to that of Robert Abernethy/Abernathy/Ebernethell of Charles City County, VA (b. cir 1632/3 Scotland, d. before 3 Feb 1685).