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Tony Lloyd, the Chairman of the Association of Police and Crime Commissioners, has welcomed calls for greater transparency in their relationships with Chief Constables and says that the "vast majority" have "developed a strong and purposeful" relationship.
"Commissioners are mindful that a good working relationship with their Chief Constable is good for force morale and will contribute to better decision-making," he said.
"The public elected Police and Crime Commissioners to hold Chief Constables to account so we welcome measures to improve transparency and enhance the confidence of the public."
The Police Federation of England and Wales said constables "should never face the possibility of being removed from their job due to political reasons or interference".
Chairman Steve Williams said: "All police officers should have the operational independence afforded to them by the office of constable - the principle that they are accountable to the law and the law alone.
"Police officers must be free to carry out their duties for the benefit of the public and to be free from compromise they must be sure their actions could not be improperly held against them in the future."
Home Affairs committee chairman Keith Vaz said it is "worrying" that Police and Crime Commissioners "seem able to side-step the statutory process for dismissing a chief constable".
Mr Vaz said: "Police and Crime Panels should make more active use of their powers to scrutinise decisions such as this.
"We will be returning to this area when we carry out our next major inquiry into Polic eand Crime Commissioners, towards the end of this year."
The Home Affairs committee said it should "not have come as any surprise" that there were a number of high-profile clashes between commissioners and chief constables following reforms that were made.
Elected Police and Crime Commissioners (PCC), which replaced existing police authorities in 41 force areas across England and Wales, were handed the power to set force budgets and hire and fire chief constables.
Just 15.1% of registered voters took part in the November 2012 PCC election - the lowest recorded level of participation at a peacetime non-local government election in Britain.
Gwent Police and Crime Commissioner (PCC) Ian Johnston said he welcomed the Home Affairs select committee's scrutiny into the role of PCCs.
He said in a statement:
A group of MPs that warned curbs on Police and Crime Commissioners are "too weak" have been particularly critical of Gwent commissioner Ian Johnston, who "persuaded" Chief Constable Carmel Napier to retire earlier this year.
Napier's retirement meant the formal process was bypassed.
The Home Affairs committee said the reasons given for the decision were "unsubstantiated by any concrete examples".
It also rounded on him for criticising the grilling he received about the move at the hands of Labour MP Chris Ruane, who represents the Vale of Clwyd constituency in North Wales.
"We were disappointed that, shortly after we took evidence from Mr Johnston, he took to Twitter to criticise a member of the committee for asking questions that he believed had been prompted by Gwent MPs, describing the proceedings as 'sad really'," the report said.
Elected police commissioners are showing a "worrying" ability to evade rules when sacking chief constables, an influential group of MPs have warned.
The Home Affairs committee claimed this was further evidence that the checks and balances on Police and Crime Commissioners (PCC) are "too weak".
Protections in place that allow police chiefs to fight their corner if they are being forced out appear to be being side-stepped, the committee said.
It claimed the early indications are that it is "very easy" for a PCC to remove a chief constable for reasons of an "insubstantial nature" and even the Home Secretary is powerless to intervene.