Aunt Petunia has become a perplexing character since Book 6. She received a Howler that we later found out was from Dumbledore, saying, “Remember my last, Petunia.” What did he mean by that? What had they been conversing about before? We know this Howler scared Petunia, so it was a serious threat. Just prior to her receiving the Howler, she also let slip that she knew what a Dementor was.
J.K. Rowling has told us in an interview that something big will be revealed about Petunia in The Deathly Hallows, but that Petunia will not be the character to perform magic later in life. What is there to learn about Petunia? Will it have a big impact on Harry?

