Ipsa says it has filed a claim at the High Court to recoup the money.
There are stories of honourable members making their frustrations very clear to Ipsa staff.
He said they would now be dealt with by the Independent Parliamentary Standards Authority (IPSA).
MPs now groan about IPSA, and press for it to exercise a lighter touch.
Guest of honour is Scott Woolveridge, IPSA's Interim Chief Executive and Accounting Officer.
Any IPSA-related motion usually provokes a chorus of complaints about the expenses regime.
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But he said the "largest challenge" so far was "the understanding of what Ipsa is about" by MPs and the public.
Ipsa took control of MPs' pay and pensions in October last year - so MPs no longer get a vote on it.
The point is that the House of Commons has found a mechanism to try and apply some control to the workings of Ipsa.
Sir George suggested Mr Griffiths should raise the issue with the liaison group, set up to iron out issues between Ipsa and MPs.
However, the IPSA, which was was set up following a public outcry over MPs' expenses, will now be in charge of overseeing all allowances.
The Speaker's Committee on Ipsa is responsible for funding the body and recently suggested how it could improve the way it handled MPs' expenses claims.
The Ipsa chairman has given evidence in recent months to two other MPs' commitees - the Speaker's Committee on Ipsa and the public accounts committee.
The MPs' expenses watchdog Ipsa is due to take control of MPs' pensions from April 2012, subject to a parliamentary procedure to allow the change.
Nor is it obvious that the new system will save money: it will be up to IPSA to set the limit for rental and other claims.
The committee's Conservative chairman Adam Afriyie asked if local media coverage was considered by Ipsa, when it considered the impact of expenses publication on MPs' reputations.
YouGov conducted online interviews with 100 MPs on Ipsa's behalf, and weighted the results slightly to represent the Commons by party, gender, year elected, and geography.
He said the first Ipsa knew about it was when he turned up at the office with a cheque to refund the amount he had been accidentally paid.
The prime minister's official spokesman said David Cameron believed it was a matter for Ipsa, when asked about the prime minister's thoughts of MPs' demands for more pay.
Sources have declined to say whether MPs' would have to make an additional 3.2% in contributions and whether any change would be made before or after Ipsa took control.
Sir Ian explained said the post was independent of the rest of Ipsa but added he was dealing with "about 100 inquiries a month" and about 40 "preliminary investigations".
Sir Ian said transparency was a vital part of Ipsa's role, but it must be "meaningful" transparency, and there were issues with how certain claims were singled out by the media.
Despite being denied a vote on the changes, MPs can make life tough for IPSA. But Harriet Harman, the leader of the House of Commons, has urged them to accept the reforms.
It looks as if MPs have started the process of bringing their expenses watchdog, Ipsa, to heel - after a bout of brinkmanship between its critics and the Government whips, last night.
In their coalition agreement, the Conservatives and Lib Dems agreed to "consult" with the Independent Parliamentary Standards Authority (Ipsa) about "how to move away from the generous final-salary pension system for MPs".
Sir Ian said he was considering publishing claims under generic headings such as "office costs" - rather than item by item - as long as Ipsa itself was able to keep itemised records.
The motion is the product of the Conservative backbencher Adam Afriyie, who has waged a long campaign, chronicled in this blog, to reform the workings of Ipsa, which remains loathed and abominated by most MPs.
During exchanges on future Commons business on 30 June 2011, Tory MP Andrew Griffiths said he mistakenly submitted two invoices with the same supplier name, date, reference and amount, which were both processed by Ipsa.
Commons Leader Sir George Young said a recent review of the Independent Parliamentary Standards Authority (Ipsa), set up in the wake of the expenses scandal, had found it needed to "improve the quality of its performance".
Labour MP Nick Raynsford, one of the committee, questioned how much Ipsa had contributed to a restoration of public trust - and how much had been due to people simply forgetting about the expenses scandal over time.
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