Six year old scary behavior

Patient: I have a 6 year old daughter who behaves very well most of the time, but at least daily displays some sort of disturbing behavior. Most concerning is her treatment of animals. I recently got her a kitten, after much talk about how to care for one correctly. She has been caught putting the kitten under cold water in the sink, putting it on top of a small playhouse outside- then shaking it until she fell off, trying to put her in a vase, and closing her inside drawers..She blatantly lies about it. When I catch her, she runs away because she knows she’s doing something she shouldn’t. If she were younger, I’d attribute this to immaturity- but she is plenty old enough to understand.I understand she has to be supervised, but I simply cannot be there all of the time- and it’s impossible to follow a 6 year old from room to room all day. I’m a single mother, doing the best I can. If this continues, we’ll have to find a new home for the kitty.Additionally, she frequently yells at me when she doesn’t get her way, talks back and acts quite aggressively towards me when she doesn’t like what she is hearing. She threw a container of candy at me today when she was told she couldn’t have any(and this was a punishment for cutting holes in her school clothes at school).I try very hard to always follow through on punishments. I speak to her calmly and create rational punishments(no tv, no swimming, etc.). When I follow through on a punishment she doesn’t like she’ll often say something like “You must want me dead. I’ll just put this bag over my head because people say that’s how you die”. I never said anything of the sort! I simply said(in a quiet tone of voice)- “No, you may not have any candy tonight”.Please help me! This is really troubling, I’m not sure if I’m somehow doing something very very wrong, or if there is something psychological going on with her? This scares me. Her father is relatively un-involved in her life. He’s on his 3rd marriage and recently had another child. I’m not sure if this could be lashing out over that? We were divorced over 3 years ago, so I’d think that would be resolved in her mind.She wears glasses, but seems to tolerate this very well. Maybe it could contribute to her frustration?Thanks for any help! This is really scaring me.