Fi is ethics of relations and whatever it involves. Ime yes, Fi types care for keeping certain image and acting certain way ie acting "ethically" in front of others. They care about how they will be perceived and don't want to cause a bad impression or project a negative image.
Edit. Fe is different imo, even when they do by default ethics of relations too (IEI has 4DFi), Fe seems more concerned by the emotional tone of relations and appropriate behavior in groups or keeping up with the group standard and thats the reason why Fe NFs care also for self image, but not particularly for acting ethically in relations but creating certain self image that will keep up within the standard.