The Independent Office for Police Conduct (IOPC) is also looking at the response to a separate report of indecent exposure three days before the kidnapping.
Wayne Couzens, 48, has pleaded guilty to the murder, rape and kidnap of Ms Everard and will be sentenced in September.
The IOPC said it was running five misconduct investigations relating to her killing and Couzens’s previous conduct, and has served notices on 12 officers from several forces who are under investigations.
One of the probes is an “investigation into alleged Kent Police failures to investigate an indecent exposure incident linked to PC Couzens in Dover in 2015”.
Couzens lived in Kent, and was serving as an officer in the Civil Nuclear Constabulary at that time.
Assistant Chief Constable Tom Richards said: “It was reported at the time that a man unknown to the complainant, who was also a man, had been spotted driving a car whilst naked from the waist down. No arrests were made.”
The IOPC is also investigating alleged failures by the Metropolitan Police to investigate two allegations of indecent exposure linked to PC Couzens in London in February 2021. Two officers are being investigated for possible misconduct.
Couzens is accused of exposing himself in a fast food restaurant on 28 February, three days before kidnapping Ms Everard and hours after arranging the hire of a Vauxhall Astra used in the crime.
The IOPC is examining the circumstances surrounding how Couzens sustained head injuries in custody on 10 and 12 March, after he was arrested on suspicion of murder. It said all officers involved were being treated as witnesses.
There is a separate investigation into allegations that a probationary Metropolitan Police constable, who later manned a cordon around a scene in the case, shared an “inappropriate graphic” with colleagues via social media. Three officers have been served with gross misconduct notices.
The IOPC is also probing allegations that officers from several different police forces breached standards of professional behaviour while sharing information linked to the prosecution of Couzens via a messaging app.
One officer has been served with a gross misconduct notice and another six have received misconduct notices.
IOPC regional director Sal Naseem said: “Our sympathies remain with the family of Sarah Everard and everyone affected by her death in such shocking circumstances.
“We share the horror that many will feel, knowing that Sarah’s killer was a police officer who had taken an oath to uphold the law and protect life and property.”
It comes weeks after an inquiry into the unsolved murder of private detective Daniel Morgan accused the Metropolitan Police of “institutional corruption”.
In April, one of its officers was convicted of membership of a neo-Nazi terrorist group, in another case that raised questions around vetting and monitoring.
The Metropolitan Police said that Couzens was not subject to any misconduct proceedings since joining the force in 2018.
He had served in the Civil Nuclear Constabulary since 2011 and previously volunteered as a special constable in Kent Police from 2006 to 2010.
Scotland Yard said it stopped paying Couzens following his initial guilty pleas to kidnap and rape, which was “as soon as legally possible”, and would formally dismiss him at an upcoming internal hearing.
Commissioner Cressida Dick said: “My thoughts and those of everyone in the Met Police are with Sarah’s loved ones. It is not possible for any of us to begin to imagine what they have been going through. I am so sorry.
“I was able to speak to them earlier today and said to them how very sorry I am for their loss and their pain and their suffering.
“All of us in the Met are sickened, angered and devastated by this man’s truly dreadful crimes. Everyone in policing feels betrayed.”
Dame Cressida, who was in court when Couzens pleaded guilty to murder on Friday, called Ms Everard a “fantastic, talented young woman with her whole life ahead of her”.
“No words can adequately express the profound sadness and anger and regret everyone in the Met what happened to Sarah,” she added.
“Today as every day our thoughts are with Sarah with her family, with her loved ones and they always will be.”
Couzens transferred into the Metropolitan Police Service (MPS), from the Civil Nuclear Constabulary, in September 2018.
His first posting was to south London, serving initially in a Safer Neighbourhood Team, before joining a response team covering the Bromley area in February 2019.
He then moved to the Parliamentary and Diplomatic Protection Command in February 2020 where his primary role was to patrol diplomatic premises, mainly embassies.
Scotland Yard said that following his arrest, a review confirmed he passed vetting processes.