Nursing woman felt strong weakness during Yom Kippur and husband commanded her to eat & drink and she did. If this is permitted or she did aveira?
Presumably she thought her husband was telling what to do in accordance with the halacha, and so the question of the aveira would be primarily on the husband. Feeling strong weakness is a vague term. Generally eating and drinking on Yom Kippur is only permissible when there is some concern of danger to life. This is usually not the case with a nursing woman, unless she became severely dehydrated, which can be life threatening. Even when something is necessary, generally we would attempt drinking small amounts [35cc] every 10 minutes or so, to avoid eating the complete amount which is liable to punishment by Torah law. Drinking sweet liquids usually circumvents the need for food. If a woman was in danger of her milk drying up we would permit her to drink the small amounts as described above.
In general a Rav should be consulted for all questions of eating on Yom Kippur, a laymen should not make this decision on his own, whether to be lenient or stringent. If the husband did err, he should do some form of teshuva, perhaps learning the laws of a sick person on Yom Kippur found in O:C 618.