Australia were 191 for seven, with Head 61 not out and tailender Mitchell Starc on eight, still 59 runs short of India's modest total of 250 at the close of another baking hot day at Adelaide Oval.
While the South Australia captain's second fifty in his third test was a boost to home fans, it merely glossed over the fault-lines snaking through the batting in Tim Paine's team.
Having wrapped up India's innings with the first ball of the day, the top order was almost as quick to blow Australia's advantage.
Aaron Finch was bowled for a duck third ball by paceman Ishant Sharma and number four Shaun Marsh's dreadful run of form continued when he dragged spinner Ravichandran Ashwin onto his stumps for two after an excruciating 19-ball innings.
Debutant opener Marcus Harris, one of three Ashwin victims, was quick to blame a "tough" drop-in wicket and worked hard to put a rosy spin on the batsmen's travails.
"It's not as easy as the old Adelaide Oval wicket," said Harris, who was caught in close for 26 after nicking onto his pads.
"We're definitely still 100 percent in the game."
Wicketkeeper-captain Paine's 20-ball five was almost as tortured as Marsh's innings and he was caught behind prodding a leaden-footed drive at Ishant.
The batsmen who did get in could not go on with it, with Usman Khawaja gloving a catch for 28 off Ashwin and recalled middle order batsman Peter Handscomb throwing away his wicket for 34 with a loose angled shot off Jasprit Bumrah.
Paine's dismissal reduced Australia to 127 for six, but Head found rare defiance in a batting partner from tailender Pat Cummins.
The pair carved out Australia's first 50-run partnership, with Head cutting a Murali Vijay delivery through the covers for a single to bring up his fifty and trigger a roar from the sun-baked terraces.
"It's nice to do a good job for the team," said Head.
"We're hanging in there, India bowled exceptionally well and put us under pressure.
"Fantastic little fight-back at the end there and sets up a really important first hour in the morning. If we can get close it's game on."
He and Cummins' stand was broken as soon as the second new ball was taken, however, when Cummins declined to play a searing Bumrah delivery and was out lbw.
Despite a slew of false shots and beaten bats, Starc and Head somehow survived the final overs to give Australia hope of mustering a total on day three.
Having lost nine wickets on day one, many of them cheaply, India were rightfully proud of their bowlers' work on day two.
"I see it as neck and neck as far as the game goes right now," said Ashwin, who conceded only 50 runs in a heavy 33-over load.
"Whoever leads first of all gets the momentum from here and obviously has an edge in this test match.
"I feel it's extremely well-poised and every run is going to be gold dust here."
(Editing by Nick Mulvenney/Amlan Chakraborty)
