Acid-free double-sided sticky tape is your friend for mounting stitching in all kinds of cards. The hard bit is getting the stitching aligned with the hole. What you can do is apply the tape around the card's hole, pin your stitching to a cork tile or something similar so it's flat, then press the card down onto the fabric once you've got it positioned right. Make sure it's stuck before you remove the pins and turn the card over though. Then use more double-sided tape to stick the backing fold in place to conceal the back of your stitching.
If you use the single fold cards, you can stick some pretty paper to the card for a backing, then fray the edges of your stitched piece before sticking it to the card.
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If you use the single fold cards, you can stick some pretty paper to the card for a backing, then fray the edges of your stitched piece before sticking it to the card.