Is it permissible for a woman to work to financially assist her husband with household expenses and building a home, given that the husband believes she will receive her sustenance while staying at home?

There is no objection to the wife helping her husband in his work, as long as it is within the Sharia guidelines, and with the husband's permission and consent. Her going out to work does not contradict reliance on Allah, and it is better for the husband to permit her, unless it affects the upbringing of the children and the care of the family. Exerting effort in seeking sustenance is commanded by Sharia and does not contradict belief in divine decree.