Sir Alan Wilson helped found the Russell Group in 1994 as vice-chancellor of the University of Leeds, but today he has concerns over "a lurking issue about the mission groups - and that is how they relate to each other".
The early atmosphere of "mutual respect" has become "more difficult to sustain in a much more competitive environment", Sir Alan said.
As director general of higher education in the Department for Education and Skills from 2004 to 2006, he argued to ministers, "successfully I think, that it wasn't helpful to talk about 'top universities' - a battle that is now lost I suspect, not for ministers in particular but more generally".
Sir Alan added: "There are considerable and complementary strengths in all parts of the sector, and it would be better if the access debate, for example, could be cast in these terms." But that, he conceded, may be "too idealistic".
Indeed the new, still more competitive higher education system could sharpen the divides between the mission groups. Universities are already jockeying for position by switching groups.
So which mission groups have the best chance of future success in lobbying? Or are the groups more useful to universities as "brands" understood beyond the sector?
Intellectual connotations
The Russell Group is already established as a brand. It is also the mission group that has developed most since its foundation.
Its earliest meetings were held in the Russell Hotel in central London, close to Universities UK, and an element of branding seems to have been in play in putting forward the name, recalled Sir Alan, who proposed it.
"I said: 'We're in the Russell Hotel, an early hero for me was Bertrand Russell, so Russell has respectable intellectual connotations - so why not?'"
A belief that Russell Group status will give a competitive edge in the new system was shown by the recent announcement that the universities of Exeter, Durham and York and Queen Mary, University of London, would ditch the 1994 Group and move over to the Russell Group, a move that officially takes place on 1 August.
Whether it is thanks to the efforts of the press or the Russell Group itself, the brand - often dubbed "the British Ivy League" - has concrete impact.
For the Department for Education under Michael Gove, which wants Russell Group universities to design A-level curricula, the group represents the UK's "leading universities".
However, the group did not necessarily have this status when it was formed. Roger Brown, professor of higher education policy at Liverpool Hope University, said the "self-declared elite" emerged from the desire of members - mainly universities with medical schools - to protect their research funding after the polytechnics became universities in 1992.
Sir Colin Campbell, former vice-chancellor of the University of Nottingham and a Russell Group founder, said "research was very important" in establishing the group. Before 1994, he said, medicine had been poorly treated in two successive research assessment exercises.
"The world is a better place for the Russell Group existing," he argued. The group had been able to explain to governments and opposition parties "the importance of research, the importance of international competitiveness being the benchmark, the importance of research-led teaching [and] the importance of having critical mass in research".
Sir Colin also said: "It is now a global brand...If you say you are from a Russell Group university, that means something if you are standing in Brazil, China or India."
But is that brand understood by students? "The very clever schools understand it," Sir Colin said. "They know when to push a student towards the Russell Group and when not. Sophisticated parents know that."
But for Sir David Watson, professor of higher education at the University of Oxford, the only genuine "break" in the sector "is between the four or five 'apex' institutions - with much larger levels of funding from all sources, and much higher research concentration - and the rest. That in itself calls into question the common interest, except nakedly political, of the larger Russell Group."
Negative briefings between the various mission groups undermine the "controlled reputational regime" that is "a source of the comparative strength of the UK higher education sector", he added.
The Russell Group has been "far and away the most successful [of the mission groups] in terms of lobbying", according to Professor Brown, pointing to government policies for higher tuition fees and the removal of number controls on the recruitment of students with grades of AAB or above at A level.
But others dispute this image of a lobbying juggernaut, saying the real power lies with universities such as Oxford and Cambridge, to whom Russell Group membership is peripheral.
And the group is not popular with David Willetts, the universities and science minister, according to some in the sector.
The minister was reportedly unhappy with the Russell Group over its perceived lobbying against his appointment of Les Ebdon, the former Million+ chair, as director of fair access.
Greater harmony
The non-Russell groups can also trumpet some recent lobbying successes: the government's reverse on the margin system and the lowering of the AAB threshold to ABB.
And GuildHE - a representative body akin to a mission group in some respects - successfully lobbied the government to ease the path to university title.
Libby Hackett, director of the University Alliance and former director of research at the Russell Group, said: "Those that shout the loudest are not always those that have the most impact with ministers."
She said the University Alliance wanted to "express the value of the whole sector, not just Alliance universities" - and produce evidence to help ensure that the sector "doesn't get a vision put upon it by a particular government" but instead steers its own future.
Andy Westwood, chief executive of GuildHE, said the looming spending review might prompt greater harmony between the groups.
"Mission groups or representative bodies...have a twofold responsibility, particularly in the comprehensive spending review. One is to your bit of the sector and the other to the sector as a whole," he argued.
All the mission groups have new or recently installed chairs - which might also help to reset relations as they prepare to battle for funding in the coming years.
Bodies of influence
One trend among the mission groups is a drift of members from Million+ (which defines itself as a thinktank) to the University Alliance.
Some of the departures reportedly stemmed from unhappiness over the supposed failure of Million+ - traditionally closer to the Labour Party - to influence coalition policies. Since 2009, Glasgow Caledonian University, Kingston University and the University of Teesside have switched from Million+ to the University Alliance, while the University of Roehampton and Southampton Solent University have also left. The Alliance is expected to announce another new member in August.
But Pam Tatlow, Million+ chief executive, said that, as a thinktank, its "affiliations have varied from year to year. We expect that." She added that the group would continue to act as a thinktank and conduct research on funding and policy changes - including how those changes affect students.
Million+ can point to lobbying successes such as the government's recent decision that all those studying for Access to Higher Education qualifications will have their further education loans written off once they complete a higher education course.
The decision of David Willetts, the universities and science minister, to appoint Les Ebdon, the former Million+ chair, as director of fair access may also indicate that the group is taken seriously by government.