Upon the death of his father, Thomas Wriothesley, the First Earl, five year old Henry Wriothesley became the Second Earl of that title on July 30, 1550. His mother, the former Jane Cheney, as a devout Catholic and remained so in spite of changes in the official religion of England. When Henry was baptized, both Henry VIII and the Princess Mary were godparents, and his mother was effectively his guardian until he came of age (in spite of a gentleman being named his official guardian), raising him in the Catholic faith.
Although he was raised Catholic, his noble family standing shielded him from the normal disabilities, fines, and other punishments of the Recusant era. He married Mary Browne, the daughter of Anthony Browne, 1st Viscount Montagu, who also remained Catholic during the succeeding Tudor changes of official religion. Browne remained as Sheriff and MP during Edward VI's reign, but took leadership positions during Mary I's reign, including travel to Rome during negotiations with Pope Julius III to reinstate Catholicism in England. Note that Browne protested the acts of Elizabeth I's first Parliament to establish the Church of England.
It was Henry Wriothesley's connection to Thomas Howard, the Fourth Duke of Norfolk and the latter's plans to marry the deposed Mary, Queen of Scots in 1569 (around the same time as the Northern Rebellion) that plunged both Wriothesley and his father-in-law into trouble with Elizabeth I. Add to that the heightened tension for Catholics in England caused by the Papal Bull of 1570, with the choice of loyalty to the Church or loyalty to the Queen, and it's no surprise that the Second Earl of Southampton spent some time in the Tower of London. This site notes that it might have been from 1571 to 1574. Then he returned to Hampshire in positions of trust--but, was suspected of having sheltered St. Edmund Campion in 1581--and died on October 4, 1581.
Henry's son Henry became the Third Earl of Southampton, Shakespeare's patron, who also spent some time in the Tower; his daughter Mary married into another Catholic noble family as the first wife of Thomas Arundell, 1st Baron Arundell of Wardour (and he spent some time in the Tower because he was a Catholic too!). After the Fourth Earl of Southamption died in 1667, the title was extinct, because Thomas Wriothesley had only daughters from three marriages.
Interestingly, I find no portrait of the Second Earl on-line at the National Portrait Gallery in London--eleven of son and ten of his grandson, but none of him! The image above is a photo of his tomb in St Peter's Church, Titchfield, Hampshire, with his mother's tomb above, from wikipedia commons. As you might surmise, St. Peter's Church was the family church, obtained from the Court of Augmentations by Thomas Wriothesley, the First Earl, upon the suppression of the Premonstratensian Abbey on December 28, 1537--the family tombs occupy the South Chapel.