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Mom may have to have a talk with her daughter even those she tends to close up. Her mom can explain to her the position she is and ask her daughter to help her to trust her and to not worry when she is away. The daughter been a teenager may believe that others (adults) do not understand her and may minimize her actions while at her father's home. Yet, mom can try and show her daughter the difficulty and possibly stress she is experiencing as a result of the non communicative father, and explain that she feels responsible for her daughter's well being because they have each other in life to trust and to respect.
Mom can find out what is prompting her daughter to all of those things away from home- opportunism, curiosity, feeling that mom is too strict, etc. Mom can explore these with her daughter and see if any compromise can be made depending on what is age appropriate behavior. Going around unsupervised near bars, etc is not; waxing armpits may be more acceptable especially if her peers are engaged in grooming of this sort.
If mom absolutely cannot talk to the father, she may try to communicate with his wife. After all, father and step mother are responsible for this girls well being when she is with them, and they would want to set some positive examples to her. Meanwhile, mom and daughter can work on establishing effective communication w. each other.