The full solution is below, with a short explanation for each group. To avoid spoilers, play the puzzle or open the hints page before reading the answer panels.
ENTER, SHIFT, ESCAPE, and SPACE all fit the category "Computer keys". They are labeled on many keyboards. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Kitchen verbs
WHISK / FOLD / SEAR / DICE
WHISK, FOLD, SEAR, and DICE all fit the category "Kitchen verbs". These are actions from a recipe. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Spicy things
PEPPER / WASABI / CHILI / GINGER
PEPPER, WASABI, CHILI, and GINGER all fit the category "Spicy things". They can wake up your taste buds. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Soft things
FLEECE / PILLOW / VELVET / MIST
FLEECE, PILLOW, VELVET, and MIST all fit the category "Soft things". They suggest a gentle texture. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.