First of all, lending your ID card to others is a very dangerous thing in itself. In case the other party does something illegal in your name, you will be in trouble. Secondly, your worry is justified, so you must investigate it clearly. Legally speaking, she took your ID card to do things. If the other party is strict, it really needs your signature authorization. After all, some things can't be solved by others with ID cards, but by myself. Of course, she may confuse you with this seemingly reasonable request, so I suggest you know the details with her before making a decision.