What are some similarities between Edmund and the author in The Lion, the Witch and the Wardrobe?

Edmund Pevensie is a perfect encapsulation of C. S. Lewis's journey to Christianity. Growing up, Lewis was an intelligent and skeptical child, quickly abandoning his parent's faith when he was a young child in boarding school. He later came to return to religion after serving in World War 1 and spending much time in discussion with fellow author J. R. R. Tolkein, among other Christian friends. However, during his stint as an atheist, he was heavily influenced by historical mythology and pagan religions—not because of their religious appeal, but more because of their psychological and historical intrigue.
Edmund is a good example of all of these things. He enters the wardrobe skeptically and is turned away from the belief in Aslan shared by his family by the White Witch, who was, herself, a representation of pagan gods and goddesses. He was tempted by his belief in his own superiority and led astray, eventually witnessing a great and destructive war. However, after the war, he returns to his family and learns to trust in Aslan, eventually becoming the most reasoned and steady supporter of Aslan and Narnia in the later books. This is clearly a parallel to Lewis, who would go on to become one of the most renowned theologians of the 20th century. Their parallels do not end there, but it is quite clear that Lewis designed Edmund to represent his own journey in this story.

I would say the key to Edmund's character arc is his temptation and redemption. If one looks at this book is an allegory for Christian theology, his story is about humanity's proclivity towards sin and the ability to seek redemption. In that respect, he is perhaps the most important of the four children thematically in terms of how he relates to the core Christian teachings—at least as far as this particular book goes. He's the one who fails the test initially, is brought back onto the right path, and emerges better for it.
In this, I'd suggest he mirrors how Lewis would have perceived his own journey from nonbeliever to devout Christian. Keep in mind, C. S. Lewis was not a lifelong Christian. He was a converted atheist, and in that respect, Edmund's own character arc might well reflect Lewis's own journey into faith.