The following information has been taken from The HP Lexicon:

While she was growing up, Petunia was disgusted by the way her parents were delighted with her sister, Lily, who was a witch and who attended Hogwarts. Petunia considered Lily to be a freak. Until the time of Lily's death, Petunia and Vernon pretended she didn't have a sister at all.
Curiously, Petunia knows more about the Wizarding world than she has let on to her family. Apparently Dumbledore has corresponded with her on occasion, and she knows more than she cares to let on about Dementors -- according to JKR, because she "overheard a conversation" between Lily and someone else. There is more to it than that, but we won't find out until Book 7 (ITV).
*So this week's discussion is on Petunia Evans-Dursley.
We know, for sure, thanks to JKR's own website that Dumbledore and Petunia have had prior correspondance. She has also stated that Petunia is not the much-talked-about person who will begin to do magic later on in his/her life.
So let us attempt to answer these question -
1) What sort of correspondance does she have with Dumbledore? Dumbledore sent her a Howler saying "Remember my last, Petunia!". We know (or guess) that Dumbledore is referring to the fact that Harry needs a family/love in order to be protected from Voldemort. Other than that, do you think that Petunia might have had other correspondance with Dumbledore about Harry?
2) Does she feel any affection/motherly-ness for Harry? Up till now, this does not seem possible, but otherwise...?
3) Does she know more than she lets on about the Wizarding World? We got a glimpse of this when she mistakingly says that Dementors guard Azkaban. She wrote it off quickly about hearing James tell Lily that - but is that really the reason? If not, what might it be? What else might she know about the Wizarding World?
4) Will she be significant later on? If so, how?
Discuss anything else about Petunia you might want to!
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