THE REASON GIVEN BY THE CHASAM SOFER FOR WOMAN/GIRLS NOT LIGHTING:
Since the original takonoh was to light OUTSIDE, and there is a rule that “KOL KEVUDAH BAS MELECH PENIMAH” A woman should not do something which is done outside for the purpose of Pirsuam because it lacks tznius. Although today many people dont light outside THE ORIGINAL TAKONO WAS NOT MADE FOR WOMEN AND GIRLS TO BY THEMSELVES BUT RATHER BE YOTZE FROM THE FATHER/HUSBAND.
The Satmer Rebbe Zatzal added to this that since part of the gezeira of the goyim of the chanuka story concerned woman “going outside” of thier family structure, (He probably was reffering to the gezeira of “tebael lehegmon techila” -ootc-) therefore at the time of the pirsum haness we purposely show the emphasis of family structure of the women/girls and have them not light independently but rather be yotze with thier father/husband.