Question: Which third cousin of Henry VIII had a great-grandfather who had a deadly rivalry with Richard Plantagenet?

Context paragraphs:
Eleanor Carey: Eleanor Carey (c. 1495 – after 1528) was the daughter of Sir Thomas Carey of Chilton Foliat and his wife Margaret Spencer.  She was the great-granddaughter of Edmund Beaufort, 2nd Duke of Somerset, which meant she was a third cousin to Henry VIII.

George Plantagenet, 1st Duke of Clarence: George Plantagenet, 1st Duke of Clarence, 1st Earl of Salisbury, 1st Earl of Warwick (21 October 144918 February 1478) was the third surviving son of Richard Plantagenet, 3rd Duke of York, and Cecily Neville, and the brother of English Kings Edward IV and Richard III.  He played an important role in the dynastic struggle between rival factions of the Plantagenets known as the Wars of the Roses.

Lady Jane Seymour: Lady Jane Seymour (c.1541 – 19 March 1561) was an influential writer during the sixteenth century in England, along with her sisters, Lady Margaret Seymour and Anne Seymour, Countess of Warwick.  Their brother was Edward Seymour, 1st Earl of Hertford.  They were the children of Edward Seymour, 1st Duke of Somerset, who from 1547 was the Lord Protector of England after the death of King Henry VIII and during the minority of Jane's first cousin, King Edward VI.  She was baptised 22 February 1541, and her godparents were Thomas Cromwell (the King's chief minister), Lady Mary (the King's daughter, at the time declared illegitimate but later to become queen) and Katherine Howard, the fifth wife of Henry VIII, and queen at the time.  Jane was thus the niece of Henry VIII's third wife, Queen Jane, whom she was probably named after.  She was the sole witness to the secret marriage of her brother Edward to Lady Catherine Grey (a potential heir to Queen Elizabeth I) in 1560.  She died a year later, aged 20, probably of tuberculosis.

Frederick Tilney: Sir Frederick Tilney (died 1445) Lord of Ashwellthorpe, Norfolk, and Boston, Lincolnshire, England, was the husband of Elizabeth Cheney, Lady Say and father of Elizabeth Tilney, Countess of Surrey.  He is a great-grandfather of Anne Boleyn, Jane Seymour, and Catherine Howard, three of the wives of King Henry VIII of England, and a great-great-grandfather to King Edward VI, the son of Henry VIII and Jane Seymour, and Queen Elizabeth I, the daughter of Henry VIII and Anne Boleyn.

Hugh Conway (Lord Treasurer): Sir Hugh Conway was created Lord Treasurer of Ireland in 1494 by King Henry VII of England.  He replaced Sir James Ormonde as Lord Treasurer of Ireland.  In 1504 he was appointed Treasurer of Calais by Henry VII ( Cal.  Pat.  Rolls Henry VIII, vol.  ii, 365).  Hugh Conway was an early supporter of Henry Tudor, Earl of Richmond (later Henry VII), was sent by Margaret Beaufort, Henry's mother to Henry, in exile in France, in 1483 with a large sum of money and encouragement to invade England through Wales and seize the crown from Richard III.  Henry did so in 1485 and Richard was killed at the Battle of Bosworth, where Hugh was present.  Hugh was appointed Keeper of the Great Wardrobe by King Henry VII on 21 September 1485, was knighted in January 1486 at the coronation of Queen Elizabeth of York, daughter of King Edward IV, and married Elizabeth Courtenay, daughter of Sir Thomas Courtenay, Earl of Devon, in 1490.  Sir Hugh Conway was appointed Sheriff of Buckinghamshire and Bedfordshire by Henry VII in 1500.  Sir Hugh was a son of John ‘ Aer Conwy Hen ’ of Bodrhyddan Hall and Constable of Rhuddlan Castle, Flintshire, Wales, by his first marriage, and descended from the ancient line of Conway (Conwy = Wales) and de Crevecouer (The Journal Of The Flintshire Historical Society (Appendix D: Conwy Pedigree (British Museum Harleian MS 1971))).  Sir Hugh's younger half-brother was Edward Conway who married the heiress of Arrow and Alcester, Warwickshire, and was great-grandfather to Edward Conway, 1st Viscount Conway of Ragley Hall, ancestor of the present family of Seymour Conways, Marquis of Hertford.

John Carey (courtier): Sir John Carey, of Plashey (ca. 1491-1552) was a courtier to King Henry VIII.  He served the king as a Groom of the Privy Chamber, and, being a descendant of Edmund Beaufort, Duke of Somerset, John Carey was a third cousin to Henry VIII.

Lisle Papers: The Lisle Papers are the correspondence received in Calais between 1533 and 1540 by Arthur Plantagenet, 1st Viscount Lisle (c.1480-1542), Lord Deputy of Calais, an illegitimate son of King Edward IV and an uncle of King Henry VIII, and by his wife, Honor Plantagenet, Viscountess Lisle (born Honor Grenville and formerly the wife of Sir John Bassett (d.1529) of Umberleigh in Devon), from several servants, courtiers, royal officials, friends, children and other relatives.  They are an important source of information on domestic life in the Tudor age and of life at the court of Henry VIII.

Edmund Beaufort, 2nd Duke of Somerset: Edmund Beaufort, 2nd Duke of Somerset, KG (1406 – 22 May 1455), sometimes styled 1st Duke of Somerset, was an English nobleman and an important figure in the Wars of the Roses and in the Hundred Years' War.  He also succeeded in the title of 4th Earl of Somerset and was created 1st Earl of Dorset and 1st Marquess of Dorset (previously held by his father and later forfeited), and Count of Mortain.  He was known for his deadly rivalry with Richard Plantagenet, 3rd Duke of York.

Baron Rokeby: Baron Rokeby, of Armagh in the County of Armagh, was a title in the Peerage of Ireland.  It was created in 1777 for The Most Rev. Dr Richard Robinson, Lord Primate of All Ireland and Lord Archbishop of Armagh, with remainder to his brothers and his father's second cousin Matthew Robinson and the heirs male of their bodies.  In 1785 he also succeeded his elder brother as 3rd Baronet according to a special remainder (see below).  Lord Rokeby never married and was succeeded in the barony and baronetcy according to the special remainders in the letters patent by his third cousin Matthew Robinson-Morris, the second Baron and fourth Baronet.  He was the son of Matthew Robinson (by his wife Elizabeth, daughter of Thomas Drake and his wife Sarah, daughter of Thomas Morris, of Mount Morris), son of Thomas Robinson, son of Sir Leonard Robinson, brother of William Robinson, great-grandfather of the first Robinson Baronet (see below) and the first Baron Rokeby.  The second Baron was an academic, politician and eccentric.  Born Matthew Robinson, he assumed by Royal licence the additional surname of Morris in 1746 on succeeding to the Mount Norris Estate through his mother.  He never married and was succeeded by his nephew Morris Robinson, the third Baron.  He was the elder son of Morris Robinson.

Jeronimo Bassano: Jeronimo Bassano was an Italian musician who is noteworthy for having been head of a family of musicians — Anthony Bassano, Jacomo, Alvise, Jasper, John and Baptista — who moved from Venice to England and the household of Henry VIII to serve the court.  Jeronimo himself never moved, however, but was listed in Venice as a "Maestro of the trumpets and shawms." Through Jeronimo's son Jacomo, who alone among his brothers did not join the recorder consort to Henry VIII, but kept his primary residence in Venice, Jeronimo is very likely a great-grandfather to the composer Giovanni Bassano.

Supporting paragraphs: Eleanor Carey Edmund Beaufort, 2nd Duke of Somerset
Answer: Eleanor Carey