Perm Hair Extensions

A: The level of manipulation possible with your hair extensions depends entirely on the type of extensions you've used. In order to be able to perm the hair extensions, they need to be 100% human hair and, ideally, hair that has not been previously treated with a hydroxide-type relaxer or colored with a metallic-based dye.
If your extensions are synthetic, they generally cannot be permed in the same way as human hair. Traditional permanent-wave solutions are designed to chemically alter the structure of human hair and are not appropriate for synthetic fibers. Attempting to chemically process synthetic extensions can result in melting and irreversible damage.
The better option is to use hair extensions that closely match the texture, color, density, and wave or curl pattern of your natural hair. Since your own hair is naturally curly and the extensions are only slightly wavy, you may find that trying to make the extensions significantly curlier is more difficult - and potentially more damaging - than starting with extensions that already have a curl pattern similar to your own.
However, if you've already paid for a head full of extensions that don't match your own hair, then you need to talk to your stylist about the extensions that were used and see if they can be permed to match your own hair. Your stylist should be able to identify whether the extensions are human hair, determine whether they have been chemically treated, and evaluate their current condition before deciding whether a perm is appropriate.
Do not skip the strand test! Even if the extensions are confirmed to be human hair, a test curl should be performed on a small amount of the actual extension hair before processing the entire head. The test can reveal whether the hair will tolerate the perm solution, how tightly it will curl, how much elasticity it has, and whether the finished texture will actually blend with your natural curls.
There's also an important difference between making slightly wavy extensions curly and simply giving them more wave. Your stylist may be able to use a perm technique and a rod size that produces a curl pattern closer to your natural hair, but the result will depend on the original texture of the extension hair. You cannot always expect chemically processed extension hair to behave exactly like naturally curly hair.
There's no way for me to say for certain what would happen to any hair extensions you already have, since I have no way of knowing precisely what kinds of extensions are used in your hair. I can warn you, however, that attempting to perm extensions that have been relaxed using a hydroxide-type product or colored with a metallic-based dye can have disastrous results, potentially leaving you with a pulpy, weak, breaking mess where the hair used to be.
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See also:
Hair extensions
Perms and perming hair
My hair is curly. Is it possible to get straight extensions?