Excerpt from biography of Thomas Nash Avery:
"In 1857 Mr. Avery was united in marriage to IMiss Eliza Carroll, and their
home was blessed with six children : Catherine, who married Charles Carman, secretary of the Clifford Brick Company; James D. ; George, who is now deceased; Grace, the wife of Frank M. Dain, of Peekskill, New York; Thomas, who also has passed away ; and Elizabeth, the wife of Dr. H. S. Cooley, of Perth Amboy. "
George does not appear in the 1870 Census with his family. 
Name Sex Age Birthplace
Thomas Avery Male 33 New York
Eliza Avery Female 32 New York
Kate Avery Female 10 New York
James Avery Male 8 New York
Grace Avery Female 6 New York
In the 1900 Census, Eliza Nash reports that she was the mother of six children, with four living.
According to census records, his family was living in Orange County, NY in 1860, in Putnam County, NY in 1870, and Westchester County, NY in 1880. I conducted a search of the surname Avery in each county on Find A Grave for memorials for George Avery and did not find any matching the information. 2/28/2018
↑ William Nelson, Ed., The New Jersey Coast in Three Centuries, History of the New Jersey Coast with Genealogical and Historic-Biographical Appendix (NY and Chicago: The Lewis Publishing Company, 1902), Monmouth Co. Library - Eastern Branch, Broad St. (Rte. 35 N), Shrewsbury, NJ 07701.
↑ "United States Census, 1870," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M8VR-RBL : 12 April 2016), Thomas Avery, New York, United States; citing p. 42, family 333, NARA microfilm publication M593 (Washington D.C.: National Archives and Records Administration, n.d.); FHL microfilm 552,576.
↑ "United States Census, 1900," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:M9FB-PSH : accessed 28 February 2018), Eliza N Avery in household of Charles Carman, Martawan Township Martawan borough, Monmouth, New Jersey, United States; citing enumeration district (ED) 129, sheet 10A, family 208, NARA microfilm publication T623 (Washington, D.C.: National Archives and Records Administration, 1972.); FHL microfilm 1,240,986.
↑ "United States Census, 1880," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MZNX-GSZ : 13 September 2017), Thomas N Avery, Cortlandt, Westchester, New York, United States; citing enumeration district ED 91, sheet 144D, NARA microfilm publication T9 (Washington D.C.: National Archives and Records Administration, n.d.), roll 0945; FHL microfilm 1,254,945.
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It may be possible to confirm family relationships with George by comparing test results with other carriers of his ancestors' Y-chromosome or mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with George: