Australia still trail by a mountainous 429 runs at Optus Stadium as India close in on 1-0 lead in the five-Test battle for the Border-Gavaskar trophy.
But Head (63 not out) has produced his highest Test score since hitting a century against West Indies in January in a rare bright spot for Australia.
Head, who recently became a father-of-two, will resume batting after lunch with fellow new dad Mitch Marsh (five not out).
After starting Monday reeling at 3-12 in chase of a record 534, Australia lost opener Usman Khawaja just seven balls into the first session.
Khawaja (four) skied an attempted pull shot off Mohammed Siraj, giving Indian keeper Rishabh Pant a simple catch.
When Head joined Steve Smith at the crease at 4-17, Australia were already tumbling towards one of their most embarrassing Test defeats.
The pair put on a 62-run stand - Australia's highest partnership of this Test - to momentarily raise spirits in the dressing room.
After a golden duck in the first innings, Smith was showing some encouraging signs he was rediscovering his best form.
But about 25 minutes before lunch, Smith (17) edged a Siraj delivery to Pant, deflating any mini-comeback for Australia.
Big wicket for India! — cricket.com.au (@cricketcomau) Siraj with a beauty! #AUSvIND pic.twitter.com/NEJykx9AvjNovember 25, 2024
The pitch is starting to deteriorate, as balls regularly keep low, but Head expertly handled the bowling attack that have caused havoc since the last session of day one.
India have obliterated Australia since midway through day one, with the expected smashing set to be in a similar realm with the disastrous 2010/11 Ashes losses and a nightmare defeat to South Africa in 2016.
It will be only India's second Test win in Perth - and first since their triumph at the WACA Ground back in 2008.
Australia last won a Border-Gavaskar series in 2014-15, with India holding the trophy since 2017, which has included winning on their past two tours to this country.
In their first innings in Perth, the Aussies made their lowest home total against India since 1981, replying to the tourists' 150 with 104.
The last time Australia recovered from a 1-0 deficit and came back to win a Test series was in the 1997 Ashes in England.
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