The multi-millionaire businessman told The Yorkshire Post that he is planning major investment in technology and people at the Huddersfield-based company.
The transaction is believed to value FMG at around £40m, generating a solid return for previous backer Spirit Capital, which invested £7m in a 40 per cent stake in 2008.
FMG is a leading independent outsourcer of fleet incident management and specialist recovery services to the fleet, leasing and insurance markets. It employs 290 people.
The business is on track to deliver core earnings of £5m on sales of £90m this year, according to Mr Cope, who is the executive chairman.
He said: “Turnover is up, profitability is up, efficiency is up and the workforce is growing again. We are actively recruiting. The business is in investment phase.”
The deal was completed in December but needed the approval of financial and legal regulators.
Mr Cope led five private equity buyouts at fleet manager Zenith between 2003 and 2014.
The company is thought to have made corporate history by driving through more MBOs than any other business in Yorkshire, if not Britain.
Mr Cope, who is 50, joined FMG in April 2014 as executive chairman after buying stakes from founder Nick Brown and Spirit Capital and has refocused the business on core activities.
The management team holds 58 per cent in the company following the latest deal.
Darren Forshaw, partner at Leeds-based Endless, said he was attracted by the quality of the management team and the future potential of the company.
He added: “We are well known for turnaround opportunities, which this is not.
“We are always looking for opportunities where there is a good underlying business, a good management team, a good opportunity to improve value, employment and grow the business.
“Those dynamics deliver good returns for us and our investors.”
Last year was a record one for Endless, which deployed £150m in capital.
Mr Forshaw said: “This year has started at the same rate as last year finished. We expect to announce two or three more deals in the coming days.”
These are understood to be in the media and communications sectors.
The Endless portfolio includes 18 companies and supports around 12,000 jobs.
The 10-year-old firm closed its fourth fund at £525m in December, taking total funds under management to £1bn.
Mr Forshaw said the portfolio is performing well and more generally business confidence is high, although “people are starting to get more concerned about the outcome of the election and probably rightly so”.
FMG specialises in providing outsourced incident management services for a range of clients including insurance companies, corporates with large car fleets and vehicle-leasing companies.
It also provides specialist roadside recovery services for the Highways Agency along with a number of police forces and local councils.
FMG has grown the fleet it manages by nearly 60 per cent over the past two years to nearly 250,000 vehicles.
CEO John Catling and the rest of the senior management team increased their shareholdings. Mr Brown remains a significant shareholder.