Cook, who walked out to a guard of honour from the India team, was soon into his stride with two trademark boundaries off successive balls from fast bowler Jasprit Bumrah.
In contrast to Moeen Ali's five-wicket haul, Ashwin struggled to get going on Saturday and finished with 1-78 in 35 overs despite his good bowling form earlier in the series.
Buttler scored 69 and put on vital lower-order partnerships with Ben Stokes and Sam Curran as England recovered from 122 for five to finish at 260 for eight at stumps on day 4.
Sharma has played a total of 86 Test for India taking 250 wickets with an economy rate of 3.25.
Jasprit Bumrah strikes twice as England struggle to reach 57 vs India at lunch on Day 1 of fourth Test
Bumrah was handed the new ball and he posed problems to the English batsmen from the very beginning.
Hardik Pandya also made an useful contribution after his maiden five-wicket haul, smashing a run-a-ball 52 with the help of seven boundaries and hit over the fence.
India played just one practice match before the series, a game curtailed to three days from the originally planned four days.
India missed a spinner at Edgbaston and a fast bowler at Lord’s. On the other hand, Ben Stokes missed out on the second Test but England didn’t miss him at all.
England and Wales Cricket Board (ECB) said in a statement that it will make an assessment of all-rounder Ben Stokes' availability after the conclusion of the trial in Bristol.
The low scores and one-sided defeat have led to a fall in rankings for most India players with off-spinner Ravichandran Ashwin being the notable exception.
India coach Ravi Shastri and captain Virat Kohli duo might face BCCI questions for debacle in England
The question doing the rounds in the BCCI corridors is why the Shastri-Kohli duo has been given a carte blanche in team management.
Virender Sehwag, Bishan Singh Bedi and VVS Laxman were among those who lamented the lack of fight shown by India in the second Test against England.
Anderson is only the second bowler, after legendary Sri Lankan spinner, Muttiah Muralitharan, to have claimed 100 Test wickets at a single venue.
India slumped to 130 all out in their second innings, with James Anderson taking his 100th Test wicket at Lord's en route to four for 23.
Anderson, England's all-time leading Test wicket-taker, removed both India openers before lunch and Broad followed up in the second session.