Praise be to Allah.
If a woman has no choice but to do wudoo’ in a place where men are passing by, then she can cover herself by having another woman stand in front of her to protect her from passing glances, or with a screen or cover held up by her sister, or she can turn her back on passers-by in such a way that men will not be able to see whatever she uncovers when doing wudoo’.
As for putting her hand into her sleeve, that will not properly accomplish the washing of the arms, because what is required is for water to flow over the part that is being washed and reach every part of it, and that is not possible if she washes her arms when they are inside the sleeves.
With regard to wiping the head, there is nothing wrong with wiping the front part of the head, then completing the wiping over the head cover, because a concession allowing a woman to wipe over her head cover was granted by a number of the scholars, especially if there is a need to do that, as has been explained previously in fatwa no. 139719.
Shaykh al-Islam Ibn Taymiyah (may Allah have mercy on him) said:
With regard to a woman wiping over a head cover that goes around her chin, two views have been narrated:
The first view is that it is not permissible, because the texts which grant the concession only apply certainly to men, and there is some uncertainty as to whether they apply to women.
The second view is that it is permissible, and this is more likely to be correct, because of the general meaning of the words, “wipe over the khuffayn [leather socks] and the head cover”. Women are included in these words along with men, just as they are included with regard to wiping over the khuffayn.
Imam Ahmad and Ibn al-Mundhir narrated from Umm Salamah, the wife of the Prophet (blessings and peace of Allah be upon him) that she used to wipe over her head cover. If she had not known that from the Messenger of Allah (blessings and peace of Allah be upon him) by hearing a statement to that effect from him, she would not have done it. End quote.
Moreover, it is permissible for a man to wipe his head from over his headgear, so it is permissible for women as for men.
And it is permissible to wear something on the head, and it is usually difficult to remove it, so it is likened to a man’s turban or head cover.
And it is more appropriate that this should be the case, because the woman’s head cover covers more than the turban or head cover of a man, and it is more difficult to remove it, and her need for it is greater than her need for the khuffayn.
End quote from Sharh al-‘Umdah, Kitaab at-Tahaarah, p. 265
We have previously described how a woman should wipe her head when doing wudoo’ when it is easy to do so, in the answer to question no. 112171
And Allah knows best.