All perfect praise be to Allah, The Lord of the Worlds. I testify that there is none worthy of worship except Allah, and that Muhammad is His slave and Messenger.
It is permissible for a woman to wear kohl in her eyes when going outside of her home on the condition that she covers her face in a way that this is not seen in her eyes as it is a kind of adornment.
Indeed, a Muslim woman is obliged to cover her adornment which she can cover if she is in a place where non-Mahram men can see her.
Therefore, it is better for her not to apply kohl when going out lest that could be seen from her unintentionally [by non-Mahram men].
Allah knows best.