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“The government is seeking ways of making public services more efficient and we’ve got a very good track record at managing sensitive places. We’re just an alternative delivery system,” he says, adding that the public doesn’t care who delivers the service, so long as it is done with quality.

Others back him up, pointing to Serco’s success in remotivating staff previously flattened by the “dead hand” of state control, and in bringing in investment where cash was scarce.

“There is a rigour to its approach to any given situation that is quite special,” says Paul Baker, managing director of Merrill Lynch, one of the firm’s advisers. “Little hierarchy, very team-orientated and extremely good problem solvers.”

Beeston describes what they do as “applied management consultancy”: not just giving opinions, but going in and applying them. “We’ve learnt to manage change. We’ve taken over 15,000 people from the public to the private sector, and done it with good industrial relations.”

But could the firm, which is building business outside Britain and across an increasing number of sectors, be stretched too far? Not if growth is managed judiciously, argues Beeston, and there are enough Serco people coming through to handle the new opportunities. Senior executives come up the same route, either running contracts, or as specialists in finance or law, then heading whole divisions.

Beeston did all three, working on finance for the firm’s European Space Agency contract (where Serco provides IT), running International Aeradio (an air-traffic control subsidiary in the United Arab Emirates), heading Serco International, becoming group finance director, then chief executive before taking on the executive chairmanship last year.

He is, according to those around the company, a different style from what’s gone before, but an effective focus for the inevitable changes the firm is going through. “As we deliver more public services,” says one board director, “we have to be more accountable.”

And that’s not just to define itself from its rivals but also because Serco employees feel that, because of its low profile, the firm isn’t given the credit it often deserves. On a wider level, too, there is a political argument to be won over PPPs and PFIs, hence Beeston’s deputy chairmanship of the CBI’s public-services strategy board.

“We want to promote the delivery of better services,” says Digby Jones, the CBI’s director-general, “and Kevin cares passionately. He wears his heart on his sleeve and he relates brilliantly to people on the shopfloor.”

That approachability was developed young. Bluff, engaging but highly ambitious, Beeston says he inherited his drive and work ethic from his parents, who moved from Ipswich to run a small garage in Great Yarmouth when he was four. Beeston and his younger brother grew up working the pumps. He could have gone to university after grammar school but was already used to paying his way, so he took a commercial-trainee job at the local Birds Eye factory instead.

He transferred down to Unilever’s Birds Eye head office in Walton-on-Thames at 21, working round the departments, before falling out with personnel over his desire to accelerate his training for an accountancy qualification. He was told even the Unilever graduate trainees couldn’t do it quicker.

“It was a red rag to a bull. I just thought, Christ, I really am a second-class citizen here. I did accelerate my studies, did pass and handed the personnel manager two envelopes: my exam pass, which she grudgingly acknowledged, and my resignation. I don’t think she was sad to see me go.”

He leapt into Serco, then RCA, in nearby Sunbury because he liked the buzz. “My career plan was to spend two years there and then go into the City, but it never happened. I feel after 18 years I’m still only halfway through.”

And he clearly loves the tough tasks. Why else, now he’s hit the top at a fast-growing firm, would he accept the invitation to become a director at his hometown football club, Ipswich, which is fast shrinking? He laughs and makes a face. He likes his sport — watching, not playing. When he’s not at Ipswich, you can usually find him at the London Irish rugby ground, where his son plays for a kids’ team. That’s just a short hop from his home in Shepperton. “It’s the centre of the UK film industry, a bit like Hollywood,” says Beeston. “At least that’s what I tell Americans. They don’t know any better.”

He laughs again. He looks anxious but happy, and so he should be. A couple of months ago, on a visit to the DLR, he confessed to a manager that his secret ambition was to be interviewed in The Sunday Times. How do I know? She e-mailed me, saying — I paraphrase — Beeston’s brilliant, Serco’s brilliant, it treats its employees brilliantly, yet nobody ever writes about it, why don’t you?

So be careful what you wish for, it may come true. Beeston looks embarrassed. It was not a PR stunt, just a cry from the heart, and the company prides itself on listening to its employees. So now he has achieved one ambition, what’s next?

“I want to take us into the FTSE 100,” he grins. Serco, with shares up 23% to 176.5p in a year is nearly in the top 150 plcs; at the current rate of growth, it won’t take long. But can the firm hold together its widely different interests as it does so? How will it maintain its family feel? Which way will the political argument go?

You can bet we’ll see more of Beeston’s bounce as he argues the toss — certainly more than we ever saw of his predecessors.