Is it permissible for her to wear kohl when going out of the house?

Every believing woman is
obliged to cover her beauty and adornment before non-mahram men, because
Allaah says (interpretation of the meaning): 

“And tell the believing women to lower their gaze (from
looking at forbidden things), and protect their private parts (from illegal
sexual acts) and not to show off their adornment except only that which is
apparent (like both eyes for necessity to see the way, or outer palms of
hands or one eye or dress like veil, gloves, headcover, apron), and to draw
their veils all over Juyoobihinna (i.e. their bodies, faces, necks and
bosoms) and not to reveal their adornment except to their husbands, or their
fathers, or their husband’s fathers, or their sons, or their husband’s sons,
or their brothers or their brother’s sons, or their sister’s sons, or their
(Muslim) women (i.e. their sisters in Islam), or the (female) slaves whom
their right hands possess, or old male servants who lack vigour, or small
children who have no sense of feminine sex. And let them not stamp their
feet so as to reveal what they hide of their adornment. And all of you beg
Allaah to forgive you all, O believers, that you may be successful”

[al-Noor 24:31] 

Adornment includes kohl, makeup, jewellery, etc. 

With regard to the words at the beginning of the verse – “And
tell the believing women to lower their gaze (from looking at forbidden
things), and protect their private parts (from illegal sexual acts) and not
to show off their adornment except only that which is apparent” – what
is meant by “that which is apparent” is the clothing, abayah (outer garment)
and headcover, and whatever appears unintentionally, because of the wind,
for example. 

Ibn Katheer (may Allaah have mercy on him) said: i.e., she
should not show anything of her adornment to non-mahrams, apart from that
which cannot be hidden. Ibn Mas’ood said: Such as the rida ‘ (cloak) and
clothes, i.e., what the Arab women used to wear of a miqna’ (a kind of outer
garment) and what appears below from the garment. There is no sin in that,
because this is something that cannot be hidden. That is like what appears
of a woman’s lower garment, and what cannot be hidden. 

End quote from Tafseer Ibn Katheer, 3/274 

Some of the scholars
interpreted the external adornment as referring to the face and hands, but
that is a less correct view, because there is a great deal of evidence that
shows that it is obligatory for a woman to cover her face. See questions no.
11774 

The scholar Muhammad
al-Ameen al-Shanqeeti (may Allaah have mercy on him) said: The more correct
of the two views in my opinion is the view of Ibn Mas’ood (may Allaah be
pleased with him), that the external adornment is that which when one looks
at it, one cannot see any part of the woman’s body. We say that this view is
more correct because it is the most cautious view, and is farthest removed
from causes of fitnah, and is purest for the hearts of men and women. It is
obvious that a woman’s face is the focus of her beauty, and seeing it is one
of the greatest means of temptation. As is well known in sharee’ah, this is
the best means of avoiding falling into something haraam. 

End quote from Adwa’ al-Bayaan, 6/200 

The basic principle is that a woman should cover her entire
face, but it is permissible for her to uncover her eyes so that she may see,
subject to the condition that showing her eyes will not lead to any fitnah
due to her wearing kohl or having wide openings in her niqaab. 

The evidence for this concession in wearing niqaab and
uncovering the eyes is the report narrated by al-Bukhaari (1838) from
‘Abd-Allaah ibn ‘Umar (may Allaah be pleased with him), according to which
the Prophet (peace and blessings of Allaah be upon him) said: “A woman
in ihraam should not wear niqaab or wear gloves.” This indicates that it is
permissible for women who are not in ihraam for Hajj or ‘Umrah to wear
niqaab. 

Abu ‘Ubayd said, describing the niqaab that was worn by the
Arabs: It is that which shows the eye-socket; they used to call it
al-waswasah and al-burqa’. Fataawa al-Lajnah al-Daa’imah, 17/171 

The reason why it is haraam to show this adornment is so as
to protect women’s chastity and honour, and to close the door to temptation
and prevent her being tempted or tempting others. Those who are sick at
heart may have hopes concerning those who show their adornment, but they
will leave the one who is modest and covered alone. 

Islam closes the doors that lead to men being tempted by
women and vice versa. Islam enjoins lowering the gaze and forbids tabarruj
(wanton display), free mixing and being alone with women. Women are warned
against going out wearing perfume or travelling without a mahram. This is
reflective of the perfection of Islam, for men by nature are affected by
women, and if this is not prevented then there will be much fitnah
(temptation and tribulation), and corruption will become widespread, as we
can see in societies that have neglected the guidelines and rulings of
sharee’ah. 

It says in Fataawa al-Lajnah al-Daa’imah (17/128):
Many women in Egypt put kohl on their eyes, and if I tell them that wearing
it for adornment is haraam, they say that it is Sunnah. Is that true? 

Answer: Using kohl is prescribed in Islam, but it is not
permissible for a woman to show any of her adornment, whether that is kohl
or anything else, to anyone other than her husband or mahrams, because
Allaah says (interpretation of the meaning): “and not to reveal their
adornment except to their husbands…” End quote. 

In conclusion: it is not permissible for a woman to wear kohl
in front of none-mahram men, because it comes under the heading of adornment
which she is required to conceal. If she is going from one house to another,
where no non-mahram will see her, then there is nothing wrong with her
wearing kohl in that case. 

And Allaah knows best.