Partial Coloring Through a Cap
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What should I do to get rid of them? Also, what should I tell her to prevent it next time?
However, even if there was no lightening stage, it's obvious that using the cap didn't allow for the color to reach sufficiently to the scalp. The better option would have been to use a foiling technique for applying the color. This generally allows for application of the color much closer to the scalp and more realistic coloring results.
You should ask your stylist to use foiling techniques for future color retouches and to correct the current hair color issues you have. If he/she cannot perform the foil technique, ask him/her to recommend a stylist who can.
Bear in mind, however, that the foil color technique is a much more labor-intensive process and is generally more expensive than partial coloring through a cap, so discuss the additional costs with your stylist.
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See also:
Hair coloring
Foiling technique
Highlighting with a cap