Q: Is it permissible to use permanent hair dye? Does it allow water to sink into the skin or does it just sit on the hair like a coating? Also if the hair dye contains alcohol, can I use it and then wash it out?
A: If the colours are colours that are used by Muslims generally and the dye itself is porous then it is permissible.
Yes, it is even better to look for an alternative that is free of alcohol.
And Allah Ta’ala (الله تعالى) knows best.