It was announced in the Daily Prophet in August. SEVERUS SNAPE CONFIRMED AS HOGWARTS HEADMASTER. I admit that I still keep a copy of the newspaper in one of my old books. This is for the simple reason that despite the fact that we were scared, and he had killed Professor Dumbledore, I personally still think he was an amazing person and I still think that being headmaster is a great achievement, but then I would say that about my uncle. It says:

“Severus Snape, long-standing Potions master at Hogwarts School of Witchcraft and Wizardry, was today appointed Headmaster in the most important of several staffing changes at the ancient school. Following the resignation of the previous Muggle Studies teacher, Alecto Carrow will take over the post while her brother, Amycus, fills the position of Defence Against the Dark Arts professor.”

Several things are wrong with this because nobody resigned. She was murdered. He never taught Defence against the Dark Arts he was to teach Dark Arts and she never taught muggle studies either. Just a varying lecture on the terrors of muggle-born wizards. At least that is what I understand from people.