I have been reviewing some discussions of a married woman's obligation to light Channukah candles. It appears that according to some, a married woman, whose husband is lighting at home does not light. I read that
when a husband sees that he will only arrive home late, he should appoint his wife to light on his behalf and he will consequently fulfill his obligation. However, if this was the situation on the first night of Chanukah, the husband will not have recited the “Shehecheyanu” blessing customarily recited on the first night. Thus, if the individual is home on the second night and will be lighting the Chanukah candles himself, he must consequently recite the “Shehecheyanu” blessing since this is his first actual lighting of the Chanukah candles.
The same article states
Nevertheless, according to the Ashkenazi custom that each member of the household lights for himself, even if the husband has traveled abroad, if he finds himself in a place where he can light, it is preferable that he light in his current location without a blessing and his wife should light at home with a blessing. If at all possible though, the husband should try to hear the blessings on the lighting from another person.
So if a man does not light on the first night, when he does light on a subsequent night, he says a shehechyanu. It seems that a woman's lighting fulfills some household obligation, but does not fully remove from him the "newness" of actual lighting.
But what about the reverse?
If a woman relies on her husband's lighting (ner ish ubeiso or ishto k'gufo as the logic) on the first night and then has cause to light on a later night, does she then say a shehechyanu? While it is her "first" lighting, she has already been included in the mitzvah through her husband's lighting and unlike the case of her husband, she need not even try to light on her own – her obligation seems more fully fulfilled.
