No, they were not sent by Henry II, they took it upon their own heads to kill Thomas Becket.
Henry II and Thomas Beckett were friends, who were clashing over their roles as King and Archbishop. Henry wanted to rule the English Church, and Becket was doing his utmost to stop this from happening. After one of Becket's move to block him, Henry was heard to say something along the lines of those listed below, we will never know what he actually said.
"Will no one rid me of this meddlesome priest?"
"Will no one rid me of this troublesome priest?"
"Who will rid me of this troublesome priest?"
"Who will rid me of this turbulent priest?"
"Will no one rid me of this turbulent priest?"
"Will no one revenge me of the injuries I have sustained from one turbulent priest?"
"Will none of the knaves eating my bread rid me of this turbulent priest?"
"What a band of loathsome vipers I have nursed in my bosom who will let their lord be insulted by this low-born cleric!"
"What miserable drones and traitors have I nourished and brought up in my household, who let their Lord be treated with such shameful contempt by a low-born cleric?"
"Who shall rid me of this troublesome priest?"
Whatever the King said, it was interpreted as a royal command, and four knights, Reginald FitzUrse, Hugh de Moreville, William de Tracy, and Richard le Breton, set out to consult the Archbishop of Canterbury. On 29 December 1170 they arrived at Canterbury. According to accounts left by the monk Gervase of Canterbury and eyewitness Edward Grim, they placed their weapons under a sycamore tree outside the cathedral and hid their mail armour under cloaks before entering to challenge Becket. The knights informed Becket he was to go to Winchester to give an account of his actions, but Becket refused. It was not until Becket refused their demands to submit to the King's will that they retrieved their weapons and rushed back inside for the killing. Becket, meanwhile, proceeded to the main hall for vespers. The four knights, carrying naked swords, caught up with him in a spot near a door to the monastic cloister, the stairs into the crypt, and the stairs leading up into the quire of the cathedral, where the monks were chanting vespers. Several contemporary accounts of what happened next exist; of particular note is that of Edward Grim, who was himself wounded in the attack.