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.
SPACE, SHIFT, ENTER, and ESCAPE 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
Things that charge
LAWYER / BATTERY / RHINO / CARD
LAWYER, BATTERY, RHINO, and CARD all fit the category "Things that charge". Charge has a different meaning for each. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.
Category
Winter words
FROST / PARKA / ICICLE / SLED
FROST, PARKA, ICICLE, and SLED 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
Things that fly
DRONE / KITE / EAGLE / JET
DRONE, KITE, EAGLE, and JET all fit the category "Things that fly". They spend time in the air. The shared link accounts for all four words, which is why the remaining board words should be tested against a different rule.