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Firstly:
For a woman to work as a hairdresser for women is permissible in principle, so long as that is not accompanied by any haraam things that usually accompany this kind of work.
See the answer to question no. 120891
Secondly:
If most of the people in the place where you choose to work are not Muslims and your work will basically be with kaafir women, then it is not permissible for you to work in the field, because all of the required guidelines on working in that field cannot be adhered to with those who do not believe that these things are haraam and do not adhere to that at all. Undoubtedly working in that field will be helping them in not covering and in making a wanton display and uncovering, and even nakedness and changing the creation of Allah. Allah, may He be exalted, says (interpretation of the meaning):
“Help you one another in Al‑Birr and At‑Taqwa (virtue, righteousness and piety); but do not help one another in sin and transgression. And fear Allaah. Verily, Allaah is Severe in punishment”
[al-Maa’idah 5:2].
Thirdly:
With regard to the issue of mixing with these women and calling them to Islam, this is a noble and important matter. However, it is not permissible to participate with them in whatever they do of sins, or to help them in that. It is not permissible to commit a certain sin for the sake of a hoped-for or imagined interest. It is possible to achieve the interest of da‘wah by other ways or in other places where it is not necessary to help them in sin.