How does Gretel portray innocence throughout the novel The Boy in the Striped Pajamas?

Perhaps, Gretel is more naïve than she is innocent as she lacks judgment and does not deduce what goes on behind the wire fences, the kind of person Lieutenant Kotler is, what transpires between her mother and the lieutenant, and what her father's role is in the German government. 
In Chapter 3, after Bruno's family arrives at "Out-With," Bruno talks to his sister about how unpleasant their new home is, and she agrees. Being three years older, she explains to her younger brother that Out-With is the name of the house in which they now will live. "Out with the people who lived here before us."

"It must have to do with the fact that he [the former commandant] did not do a very good job, and someone said 'out with him and let's get a man in here who can do it right,'" (Ch.3) she tells Bruno.

As they talk, Bruno mentions that the children do not look very good, either. Gretel has no idea what he means, so Bruno invites her to look out his window. When she does, Gretel does not like what she sees. There are wire and sharp spikes all around the fence. While Gretel wonders why anyone would build such "a nasty place," she concludes that it must be the countryside where food is raised somewhere. When Bruno questions her, she tells her brother that she learned in school about the countryside where there is so much land. So, this is where they must be. Then, she "shivers and turns away" and tells her brother that she is going back to her dolls in her room where the view is much nicer.
Further, Gretel enjoys feeling older as Lt. Kotler flirts with her; as a result, she naively believes that he is nice because she lets her emotions rule her. In fact, she does not really assess what a cruel, sadistic young man he is. When Gretel is with Lt. Kotler, she "...laughed hysterically and twirled her hair around her fingers...." (Ch.9)
Gretel's mother talks privately with Lt. Kotler, calling him affectionate terms, but Gretel does not seem to notice, while Bruno catches her.

"Oh, Kurt, precious, you're still here," said Mother....I have a little free time now if--Oh! Bruno! What are you doing here?" (Ch. 15)

Of course, Gretel knows nothing of mother's "afternoon naps" and is also innocent of why Lt. Kotler is transferred. Also, she does not know what lice is when she finds an egg in her hair. Later, when her father finally decides that the children must depart from "Out-With," but it is mainly because Mother wishes to return to Berlin, Gretel is unaware of this, as well.