Is it permissible for one's wife to send photographs without a head covering to complete residency procedures in France, knowing that refusing to do so may lead to the husband losing an important job and training opportunity?

It is obligatory for a Muslim woman to adhere to her hijab (definition), and she is not permitted to expose any part of her body except for the face and hands, according to the more probable opinion. If the passport administration stipulates a photographic image of the wife without hijab, and if the intention is only to expose the face, then there is no harm in that due to the necessity of residence. However, if the intention is to expose the head, which is 'awrah (private parts) by consensus, then it is not permissible except in a case of extreme and unavoidable necessity, because necessities permit the forbidden.