Nowadays, women are attracted to working outside the home. As they are being educated, they don't want to live as a housewife. It is true that some of the women are in need of job, but most of them want to be self-dependent. Sometimes, they even don't care about their husband. So, I want to know what is the husband's role in this case? What is his right? What can women do with and without the permission of her husband?
A woman can work from home, and interact with the outside world by means such as by using an agent to act on her behalf, or by sending a letter, or by email, or by a voice call, or by using a remote freelancing website, or by using using any other collaboration tools etc. This is permissible.
As for going out of the house to work, then in principle it is not permitted, because Islam teaches women to observe hijab and to remain at their homes.
وقرن في بيوتكن ولا تبرجن تبرج الجاهلية الأولى
And abide in your houses and do not display yourselves as [was] the display of the former times of ignorance.
المرأة عورة، وإنها إذا خرجت استشرفها الشيطان، وإنها لا تكون إلى وجه الله أقرب منها في قعر بيتها
Woman is ‘awrah (that which must be concealed), and when she goes out the Shaytaan’s hopes are raised. She is never closer to Allaah than in the inner part of her house.
Especially, a woman is not permitted to leave her house without the permission of her husband, there is Ijma' on this:
مسائل الإجماع والاتفاق في باب ... وعلى أنه يجب على الزوجة طاعة زوجها وملازمة المسكن، وعلى أن له منعها من الخروج
The matters of consensus and agreement in this topic are ...
It is obligatory on the wife to obey her husband and to remain in her house, and it is the right of the husband to forbid her from going out
Some of the jurists have explicitly specified the husband's right to forbid his wife from employment (ijarah):
وله منعها من إجارة نفسها
The husband has the right to forbid her from giving herself for hire
There are exceptions to this when there is a necessity. For example the exceptions relevant to earning are where the husband or family can not provide enough sustenance. Or when the woman needs to go out to fulfill a personal obligation or a communal obligation such as being a midwife.
Further there are conditions for a woman working such as permission of her husband, observance of hijab and abstinence from adornments, segregation from men, traveling with a mahram when required etc. You can see e.g. https://islamqa.info/en/answers/106815/