He is sick and wiping his head will harm him. How should he do wudu; can he put something on his head and wipe over it?

Firstly:

It is permissible to wipe over the turban, because of the report narrated by al-Bukhaari (205) from ‘Amr ibn Umayyah, who said: I saw the Prophet (blessings and peace of Allah be upon him) wipe over his turban and khuffayn (leather slippers).

The view that it is permissible to wipe over the turban is the view of Imam Ahmad.

See the answer to question no. 129557 .

It is not valid to wipe over the kufi (white skullcap) or other kinds of hats or caps. See the answer to question no. 139719 .

Secondly:

If there is a plaster cast or dressing on your head, then you can wipe over it.

That is proven from ‘Abdullah ibn ‘Umar (may Allah be pleased with him).

Al-Bayhaqi (may Allah have mercy on him) said: There is no report concerning that from the Prophet (blessings and peace of Allah be upon him)… Rather there are the view of the fuqaha’ among the Taabi‘een and those who came after them, along with what we have narrated from Ibn ‘Umar.

He narrated with his isnaad that Ibn ‘Umar (may Allah be pleased with him) did wudu when his hand was bandaged, so he wiped over it and over the bandage, and he washed everything else. He said: This is soundly narrated from Ibn ‘Umar."(Al-Majmoo‘  2/368).

Thirdly:

If your head is bare, but will be harmed by wiping it, then you should wash all the other parts that are to be washed in wudu, and do tayammum instead of of wiping your head. It is not valid for you to put something over your head then wipe over it, unless it is a turban that you are wearing, or a dressing which may result in harm if it is removed.

It says in Kashshaaf al-Qinaa‘ (1/165): If part of his body is wounded and the like, such as if there is a sore or ulcer, and he will be harmed if he washes it or wipes it with water, he may do tayammum for it, i.e., for the part that is wounded and the like…

If he is able to wipe over the part that is wounded and the like with water, then he must do that and that will be sufficient, because washing is enjoined, and wiping is like a lesser form of washing; that is like the case of one who is unable to bow and prostrate, but is able to make gestures. End quote.

Shaykh Ibn Baz said: If he has a dressing, he may wipe over it, and if the wound is uncovered, he should do tayammum instead. End quote.

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in ash-Sharh al-Mumti‘ (1/169): The scholars (may Allah have mercy on them) said: Wounds and the like are either uncovered or covered.

If the wound is uncovered, then it must be washed with water. If it is not possible to wash it with water, then the person should wipe over the wound. If it is not possible to wipe over it, then he should do tayammum. This is the order of preference.

If it is covered with something that is (medically) appropriate to his situation, then he has no choice but to wipe over it. If wiping will cause him harm even though it is covered, then he must turn to the option of doing tayammum, as in the case where it is uncovered. This is what was stated by the fuqaha’ (may Allah have mercy on them). End quote.

Fourthly:

If someone does tayammum for a wound and the like, he may do tayammum before or after doing wudu. The Hanbalis regard it as obligatory to follow the same order as wudu, by doing tayammum at the point where one would ordinarily wipe the head.

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said:  The statement that the one who has a wound may do tayammum for it and wash the rest refers to the case when he has a wound on some part of his body [that is washed in wudu], meaning a wound that would be harmed by water. He should do tayammum for the wounded part, and wash all the other parts that are to be washed in wudu. With regard to doing tayammum in the case of the wound, it is not stipulated that water be unavailable, so there is nothing wrong with doing tayammum when water is available in this case.

The apparent meaning of the author’s words “he should do tayammum instead” is that tayammum must be done in place of washing the injured limb, because it is stipulated that the parts of wudu be done in a particular order. But if the issue is doing ghusl from janaabah when there is a wound, it is permissible to do tayammum before doing ghusl, or immediately afterwards, or after some time has passed.

This is our view, because they thought that in the case of ghusl, it is not stipulated that the actions be done in a particular order or straight after one another.

If the wound is in the hand, then you must wash your face first, then do tayammum, then wipe your head, then wash your feet.

In this case, you must have a towel with you, so that you can dry your face and hands, because it is stipulated that the soil or sand used for tayammum must be dusty and dry, and if there is water on your face, tayammum will not be valid.

Some of the scholars said that it is not stipulated that the actions be done in a particular order or straight after one another, as in the case of major impurity.

Based on that, it is permissible to do tayammum before or after wudu, either immediately afterwards or some time afterwards.

This is what people do nowadays, and it is the correct view. It was favoured by al-Muwaffaq, al-Majd, and Shaykh al-Islam Ibn Taymiyah, who regarded it as sound in Tasheeh al-Furoo‘."(Ash-Sharh al-Mumti‘  1/383).

And Allah knows best.