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.
PARKA, FROST, SLED, and ICICLE all fit the category "Winter words". Cold weather brings these out. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Fast things
ARROW / SPRINTER / FALCON / COMET
ARROW, SPRINTER, FALCON, and COMET all fit the category "Fast things". Speed is the common thread. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Things with strings
KITE / PUPPET / VIOLIN / HOODIE
KITE, PUPPET, VIOLIN, and HOODIE all fit the category "Things with strings". A string is part of the object or how it works. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Desk items
NOTEBOOK / PENCIL / LAMP / STAPLER
NOTEBOOK, PENCIL, LAMP, and STAPLER all fit the category "Desk items". They belong within arm's reach at work. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.