Is it permissible for a woman to work on Friday, and is it permissible to keep a shop open during Friday prayer if a woman is present in it?

It is permissible for a woman, a slave, a traveler, a sick person, and an underage child to sell during the time of the Friday prayer and sermon, provided that both the seller and the buyer are not obligated to attend Friday prayer, or that one party is not addressed by it while the other is. In the latter case, it is forbidden for the one who is addressed and disliked for the one who is not. Based on this, it is permissible for your husband to appoint his sister to sell in his place during the time of the Friday prayer, provided that she only sells to those who are not obligated to attend Friday prayer.