Australia all out for 349, set England 435 to win 3rd Ashes Test
The hosts resumed day four at Adelaide Oval at 271-4 with Head on 142 and first-innings century-maker Carey not out 52.
They shared a 162-run partnership before Head was caught by Zak Crawley in the deep, with Carey departing soon after to a slip catch.
England quickly wrapped up the tail with Josh Tongue the pick of the bowlers with 4-70.
The tourists face a huge task to win the Test and keep the series alive.
No team has ever chased down more than 316 at the ground, while the biggest successful run chase ever was 418 by the West Indies against Australia at St. Johns in 2003.
Australia lead the five-Test series 2-0 and will retain the Ashes if they win or draw, after back-to-back eight-wicket thumpings in Perth and Brisbane.