The Vernet Family numbers amongst it many famous painters of the eighteenth and nineteenth centuries. In "The Adventure of the Greek Interpreter", Sherlock Holmes states that one of these (although he does not specify which) was his grandmother's brother.
Antoine Vernet (1689-1753)
Antoine Vernet was a decorative painter in Avignon, France.
Claude Joseph Vernet (1714-1789)
Claude Joseph Vernet travelled to Rome during his teens to study art. Here he met and married an Englishwoman, Virginia Cecilia Parker, whose father was a captain in (and according to some sources, later the Commander-in-Chief of) the maritime forces of the Papal States. He was later recalled to France at the command of the royal family, who sought him as a member of the Academy. In this connection he became caught up in the French Revolution, and his daughter subsequently went to the guillotine.
(It may be worth noting that Wold Newton researcher and superhero specialist Al Schroeder has identified John Parker as an ancestor of Spider-Man).
Antoine Charles Horace 'Carle' Vernet, known as 'English' Vernet (1758-1835)
Carle Vernet was feted as a painter under the Directory, then under Napoleon, and under the restored Louis XVIII. An Anglophile, he introduced English-style horce racing into France.
In her article "Name of a Thousand Blue Demons" (in Myths for the Modern Age, ed. Win Scott Eckert), Wold Newton scholar Cheryl L. Huttner reveals that Carle Vernet had an affair with Philippa Delagardie, and that the relationship produced a son whom they Christened Carle after his father's nickname. Carle was the grandfather of both Hercule Poirot and Jules de Grandin.
Horace Vernet (1789-1863)
Horace Vernet painted for the restored French monarchy, King Louis-Philippe being a particular patron, and later for the restored French imperium under Napoleon III, whose armies he accompanied to the Crimea as a war artist.
Violete Vernet (Sherlockian)
According to William S. Baring-Gould in Sherlock Holmes of Baker Street and Michael Harrison in The World of Sherlock Holmes, Violete Vernet was a daughter of Carle Vernet. She married Sir Edward Sherrinford, and one of their remarkable daughters, named Violet, married Siger Holmes and accordingly became the mother of Sherlock Holmes.
? Vernet (Farmerian)
In Wold Newton circles, it is generally agreed that it was Siger's father, Mycroft Holmes, who married a Vernet. According Cheryl L. Huttner's article, her name was Violet, and she was the illegitimate daughter of Carle Vernet and Philippa Drummond. Also in Myths for the Modern Age is Brad Mengel's "Watching the Detectives, or The Sherlock Holmes Family Tree" conversely follows Mona Morstein's novel The Childhood of Sherlock Holmes in identifying Miss Vernet's name as Camille Francoise Josephine Lecomte-Vernet (but rejecting once again the idea that she was Holmes's maternal grandmother).
Dr. Horace Verner
In The Norwood Builder, it is stated that Sherlock Holmes secretly advanced his cousin, a Dr. Verner, money to buy Dr. John H. Watson's medical practice. It is generally agreed that the name "Verner" is an Anglicisation or corruption of "Vernet". A short story, "The Anomaly of the Empty Man" by Anthony Boucher, reveals that Dr. Verner later became a Fortean detective of sorts in his own right. It also reveals that he is the great-nephew of Brigadier Gerard.