Cant get on with my sister....?

I just really cant get on with my sister....shes soon to be 6,but shes constantly making smart comments and using really nasty sarcasm (which she must get from school) or shes preaching God (my stepdad brainwashed her into it) and so earning brownie points with my mum. Whenever we do something together,( i rally try my best to engage with her well) she allways storms off after about 30-60 minutes...(sometimes its my fault,but usually she just gets in a huff).I know this sounds rediculous cos shes only 6, but my mum keeps nagging me about it and when she questioned me further and i told her how it is she said that shed do the same thing to my children. So i said id never let her see my children then, and she said shed kick me out.(Jokingly). We then had a bit of a row and she told me to go away, when i was going to clear the table. I know its probably my fault, and im probably a nutcase or something, but we cant afford to see a psychiatrist...Anyone know anything to help me accept her?