It seems permissible as you are not doing it while making wudu. But if you believe that making your neck wet before or after wudu is preferable then it is b’idah.

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This does not break your wudu. Kindly refer to the following link(s): https://www.youtube.com/watch?v=iED3-xzW6A8

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For a woman, she should wipe from the roots of her hair on the forehead until the nape of your head. One time is sufficient but sunnah is to do it back and forth.

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