Carter, Jake Carpenter, Brian Carberry, Elwyn Lopez, Nick Valencia and Tina Burnside contributed to this report.
Hogg trapped Lumb lbw and Keedy removed Carberry and Benham to leave the visitors wobbling on 71-4.
CNN's Jake Carpenter, Brian Carberry, Elwyn Lopez, Nick Valencia and Tina Burnside also contributed to this report.
Michael Carberry (78) and Sean Ervine (54) looked in good touch at the crease as they led Hampshire to 159-3.
CNN's Jake Carpenter, Brian Carberry, Elwyn Lopez, Nick Valencia, Dana Ford and Tina Burnside also contributed to this report.
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Neil Carberry, the CBI's director for employment and skills said this was a "big step in the right direction".
James Adams, in his first Championship game since May, shared an opening 47 in 18 overs with Michael Carberry.
Former hairdresser Mr Carberry said he went to the job centre a few weeks ago to discuss his options.
Before the close Hampshire made 13-1 off two overs with Michael Carberry caught behind off the day's final ball.
Paul Carberry briefly looked threatening on Watson Lake turning into the straight.
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Hampshire were 46-0 by the close of play, with James Adams 17 not out and Michael Carberry 21 not out.
Michael Carberry was out to the third ball he had faced for the second time in the match without scoring.
The opening day of the four-day Irish National Hunt Festival at Punchestown proved a happy one for the Carberry family.
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Michael Carberry (50) and James Vince (44) looked to have put the home side in a position of strength at 168-3.
He said an adviser also said he would seek legal advice about what contact Mr Carberry could have with other staff.
CNN's Jake Carpenter, Dana Ford and Brian Carberry contributed to this report.
With the ball moving about under cloud cover and half the side gone for 183, Hampshire were in danger of wasting Carberry's enterprise.
But the visitors ended on a positive note with Jimmy Ormond claiming the wicket of Carberry from the final ball of the day.
Michael Carberry had earlier set the tone for the visitors with 77 before becoming one of Andrew Miller's three victims on his Championship debut.
Jayce Carberry said staff in Maidstone made him declare his condition within earshot of others, and told him he had to see a disability adviser.
Hampshire started badly when Chapple struck to remove opener Jimmy Adams for just nine before Michael Carberry and Michael Lumb moved the score on to 50.
After some uncertainty with the bat on Sunday, and the cheap dismissals of Cook, Michael Carberry and Kevin Pietersen, England were immediately in control of affairs on Monday.
Hampshire made a terrible start to their innings when Michael Carberry fell third ball to Chapple, before Brown and John Crawley put on 48 for the second wicket.
Hampshire's response began in the worst possible fashion as openers Michael Brown and Michael Carberry returned to the pavilion without score as Mahmood caused havoc with the ball.
CNN's Dana Ford and Brian Carberry contributed to this report.
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Michael Carberry was lbw to Nadif Chowdhury for 26 but Compton and Matt Prior added 74 for the fourth wicket before Prior holed out in the deep for 37.
Phil Mustard was strong on both sides of the wicket and motored on despite the departure of Michael Di Venuto, who was brilliantly caught by Michael Carberry at mid-wicket.
Peter Trego made the breakthrough with the wicket of Carberry lbw, then deceived Adams with a slower ball in the 26th over, before taking a one-handed return catch diving forward.
Their second innings then got off to the worst possible start with the out-of-form Michael Carberry being caught behind for one off Lewry for the second time in the match.
And his spin team-mate Tredwell, selected ahead of Michael Carberry as England went with five bowlers, was pleased to make his presence felt on the tour - for the second time.
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