Tea Australia 260 for 8 (Cummins 10*, Lyon 4*) lead West Indies 188 by 72 runs
The visitors chipped away during the morning session and when Alex Carey was caught behind, Australia still trailed by 20. But Head sped into the 90s with a flurry of runs against Shamar Joseph and brought up his century from 122 balls when he dispatched Gudakesh Motie through the off side.
It was his first Test century since the World Test Championship final against India in June and the first time he had passed 50 since the Headingley encounter in the Ashes. He had entered in just the second over of the day and dominated on a different level to any other batter in the game. After reaching his century, Head flicked the switch and carted Shamar Joseph for consecutive sixes over the leg side.
Head and Mitchell Starc added 54 for the seventh wicket to take Australia ahead. After Starc had fended Shamar Joseph to short leg, given him a fourth wicket, Pat Cummins provided support in another useful stand until Head picked out deep midwicket aiming for another six.
West Indies enjoyed an excellent morning to keep themselves in contention. Shamar Joseph added to his first-day success early on the second when he had Cameron Green caught behind, then Greaves broke a threatening stand between Head and Usman Khawaja when the latter was taken at slip for his first Test wicket.
On a surface that was proving trickier to bat than expected, Mitchell Marsh was given a working over around off stump before being superbly caught by Greaves, standing very close at third slip and wearing a helmet.
The early signs had been that it could be Green’s day, as he resumes his Test career at No. 4, but after a brace of early boundaries off Shamar Joseph he nicked a good-length delivery that left him a fraction to send the bowler in another sprint celebration across the Adelaide Oval outfield the morning after he became a story around the world.
Shamar and Alzarri Joseph maintained early pressure on Khawaja and Head, the former happy to bide his time while the latter was keen to be aggressive. West Indies quickly went to a short-ball plan against Head although the surface did not quite have the pace in it to be hugely problematic.
Head went past 3000 Test runs with a lofted drive over the covers as the partnership started to dominate. Khawaja was shaping up well, putting a couple of pulls away strongly through midwicket, while Head also deposited Greaves into the stands over square leg.
But just as concerns were growing as to whether West Indies could maintain the pressure, the medium pace of Greaves lured Khawaja into driving a wide delivery and the edge was well held at second slip.
Marsh struggled early in his innings, multiple times edging short of the cordon, and had 1 off 22 balls before he laced a cover drive off Shamar Joseph. However, it wasn’t a sign of a change in fortune. After another edge fell short of third slip, Greaves moved up even closer, and then brilliantly held the opportunity off Kemar Roach.
At 129 for 5, the game was in the balance but by tea it was very much advantage Australia.
Andrew McGlashan is a deputy editor at ESPNcricinfo