Yvette Cooper, Labour MP for Pontefract and Castleford, said people remained "furious" about the decision.
In the Commons, Labour's Yvette Cooper welcomed the proposals but said they could have gone further.
BBC: Police corruption: Gifts and second jobs to go on register
But senior figures, including Ed Balls, Yvette Cooper and Douglas Alexander, retain their existing positions.
And Labour shadow home secretary Yvette Cooper said agreed, adding the plans needed to be reconsidered.
Public health minister Yvette Cooper said: ""We are delighted with the outcome of this judgement.
Chief Secretary to the Treasury Yvette Cooper told the Andrew Marr Show she "completely disagreed" with Sir John's analysis.
But Shadow Home Secretary Yvette Cooper said the government was "failing badly on its own immigration target".
But Labour's Yvette Cooper said cuts were making it harder for officers to "catch criminals and deliver justice".
Work and Pensions Secretary Yvette Cooper agreed it was "essential" that single parents got the support they needed.
Shadow home secretary Yvette Cooper said it was "a matter of grave concern".
Her shadow, Yvette Cooper, made it clear she wanted to see "leadership" from both the Met and the government.
Shadow home secretary Yvette Cooper said MPs should "of course" not prejudice the police investigation and any potential trials.
Mr Miliband's married Labour colleagues Ed Balls and Yvette Cooper go for Elvis Presley's Can't Help Falling in Love.
Her old post has been given to Yvette Cooper, who is married to Ed Balls, the chancellor's economic adviser.
Mr Byrne's letter acknowledged a recent "cordial" meeting between the SNP minister and UK ministers Yvette Cooper and Jim Murphy.
More than 20 Labour MPs - led by shadow home secretary Yvette Cooper - have called on the government to reverse planned police cuts.
BBC: England riots: Cameron says police admit to wrong tactics
In a speech later today, shadow home secretary Yvette Cooper is expected to set out the Labour Party's new approach to immigration.
Shadow home secretary Yvette Cooper accused the government of rushing out the plans to avoid further bad headlines over this week's Budget.
At Home Office questions on 21 May 2012, shadow home secretary Yvette Cooper complained that border controls had been "downgraded hundreds of times".
Shadow Home Secretary Yvette Cooper criticised Surrey Police for ignoring guidelines and failing to offer Mrs Andrade counselling before the trial was over.
BBC: Surrey Police orders review of child sex investigations
The shadow home secretary Yvette Cooper accused Mrs May of a "blatant political pitch" to right-wing Tories, disillusioned with the prime minister's leadership.
Former cabinet ministers Alan Johnson, Tessa Jowell and Yvette Cooper have also entered the ballot for 19 posts - elected by Labour MPs.
Shadow home secretary Yvette Cooper said MPs had "an obligation" to protect the public, especially children and other vulnerable groups, from sex offenders.
But Chief Secretary to the Treasury Yvette Cooper told the BBC the Conservatives were out of step with the rest of the world.
News of the proposal had drawn criticism from patients and MPs including Yvette Cooper, shadow home secretary and Labour MP for Pontefract and Castleford.
Shadow home secretary Yvette Cooper said the government had "put chief constables in an impossible position with a 20% front-loaded cut to their budgets".
Housing Minister Yvette Cooper told the BBC's Today programme that the government was working hard to try and alleviate the shortage of affordable housing.
Yvette Cooper said that "confusion and chaos has turned into farce" and asked why ministers did not "pick up the phone to sort it out".
Shadow home secretary, Labour MP Yvette Cooper, criticised Mrs May's "overblown" strategy in dealing with Abu Qatada, and said "serious questions" needed to be answered.
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