Why did the Queen send away the Duchess after having her brought to the croquet?

In Chapter VIII of Alice's Adventures in Wonderland, Alice finds herself in the garden where the Queen of Hearts comes to play croquet. Alice finds herself marching in a procession next to the White Rabbit. She asks him where the Duchess is, and he explains that she is "under sentence of execution" for boxing the Queen's ears. Evidently, the Duchess had arrived late, causing the Queen to make a remark that angered the Duchess, which isn't surprising considering how hot-headed the Duchess was when Alice met her.
Before long, the Cheshire Cat appears at the croquet game—or its head does, at any rate. As Alice is having a conversation with the cat's head, the King interrupts. Soon he accuses the cat of being impertinent and calls upon the Queen to have the cat "removed." The Queen's solution is, of course, to behead the cat, but since it is already only a head, the executioner, King, and Queen argue whether that's possible. They ask for Alice to settle the argument, but she defers, saying the cat belongs to the Duchess, so they should ask her. The Queen sends the executioner to bring the Duchess out of prison and to her, but by the time he returns with the Duchess, the cat's head has disappeared. The Duchess and Alice then have a stroll and a conversation, and Alice is surprised that the Duchess is no longer angry. When the Queen confronts the Duchess, she threatens to behead her again, and the Duchess flees.
Thus the Queen doesn't exactly send the Duchess away after calling her—she merely threatens to behead her, a sentence she had already delivered, which resulted in the Duchess's imprisonment. However, the reason the Duchess was summoned—the cat—is no longer a concern because the cat has disappeared, so the Queen no longer needs the Duchess to explain the cat. The scene emphasizes the Queen's whims but also shows that her ruthless sentences are never actually carried out and that her prisoners go free. By the end of the croquet game, all the players except Alice, the King, and the Queen have been taken into custody by the soldiers and are sentenced to death. But as Alice walks away with the Queen to meet the Mock Turtle, the King pardons them all, to Alice's great relief.