The 24-year-old former Liverpool trainee had initially been offered a new two-year-contract at Brunton Park.
Mr Brunton said he was confident they could "achieve the priorities set by the Authority".
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Hartlepool's 1-0 win at Brunton Park brought increased calls for Ward's dismissal from sections of the crowd.
Abbott was appointed manager at Brunton Park in October of last year following the departure of John Ward.
Deputy Chief Constable Neil Brunton said he welcomed the Authority's "continued focus on tackling serious crime and road collisions".
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Graham, 23, will continue talks with the Cumbrians this week after stalling on a new deal at Brunton Park.
Abbott expressed sympathy for the departed Ward after 13 months working as assistant manager to the 57-year-old at Brunton Park.
The Cumbrians are also hoping to add a meeting with Newcastle United to their Brunton Park itinerary in early August.
Carlisle twice surrendered the lead in an entertaining game at Brunton Park.
Collin, 24, will train twice a week at Brunton Park with a view to a permanent switch to the League One club.
"This isn't just a naval asset, this is a joint asset for the country and the government to use, " Rear Adm Brunton said.
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Dublin-born McGill, 28, started his career with Sunderland before a four-year spell at Brunton Park where he made 132 appearances, scoring 17 times.
Rear Adm Steve Brunton presented Leading Hand Claire Butler with the first "cap tally" band for her hat with the carrier's name on it.
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Allen, who is part of a four-man team of owners at Brunton Park, has admitted that the team's current predicament has taken him by surprise.
The former England youth captain was a product of Liverpool's academy and made eight first-team appearances for the Reds before his move to Brunton Park.
The Cumbrians have now won only once in their last nine games and have failed to record a win at Brunton Park in their last seven.
Ward, 57, leaves Brunton Park following a run of nine defeats in their last 10 League One games, the latest being Saturday's 3-0 defeat by Stockport.
Birtley-born striker Madine progressed through the club's youth ranks to earn a professional deal at Brunton Park, and made his debut against Nottingham Forest in March 2008.
Both players will be out of contract at the end of the season but have so far failed to agree new contracts to stay at Brunton Park.
Abbott said he was conscious of the dip in attendances over the past year and would do all he can to get the crowds back to Brunton Park.
"What Guan did at the Masters, in my view, was very, very dangerous, " said Henry Brunton, the former national golf coach of Canada and an expert on golf development.
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The Cumbrians were able to bring in goalkeeper Tim Krul on loan from Newcastle last season while left-back Michael Liddle spent a three-month spell at Brunton Park from Sunderland.
Abbott, who released six first-teamers earlier this week, will now turn his attention to strengthening his squad although there is set to be few arrivals into Brunton Park this summer.
The signing will be a blow to Carlisle boss Gregg Abbott, who revealed to BBC Radio Cumbria he had been in talks with the forward about a move to Brunton Park.
"I and others like me feel his death so keenly in part because we aspire to be the misfit, the wunderkind, the maker that he was, " wrote blogger David Brunton.
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Graham, 23, scored 40 goals in just over 100 appearances for United since joining from Middlesbrough in 2007 but turned down the offer of a new deal at Brunton Park this summer.
But Dr Brunton's results suggest that when predators have more dimensions of action than prey as is the case when they are flying and the prey are on the ground, the reverse is true.
But Carlisle had a home record matching the 17 wins achieved by Premier League champions Manchester United and the Brunton Park side had the first chance as an off-balance Grant Smith shot wide.
The two sides will now meet in the second leg at Brunton Park on Tuesday, 9 February to decide who will meet either MK Dons or Southampton in the final on Sunday, 28 March.
Taylor, 24, suffered a serious shoulder injury in July's pre-season friendly with Carlisle United at Brunton Park, and following his return to fitness has struggled to force his way back into the first XI at St James' Park.
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