Jake Davison was deprived of the weapon and its permit in December after he was blamed for attack, the Independent Office for Police Conduct (IOPC) says.
It is to inspect why Devon and Cornwall Police chose to return them in July.
In the interim, many individuals went to a vigil on Friday evening near where the shootings occurred.
Davison, 22, shot his kid mother Maxine Davison, otherwise called Maxine Chapman, at a house in Biddick Drive before he went into the road and shot dead Sophie Martyn, matured three, and her dad Lee Martyn, matured 43.
He killed Stephen Washington, 59, in a close by park, prior to shooting Kate Shepherd, 66, on Henderson Place. She later passed on at Derriford Hospital.
Davison additionally pointed and took shots at two nearby inhabitants – a man matured 33 and a 53-year-elderly person – who are known to one another, in Biddick Drive.
His assault, in Keyham, endured around six minutes before he turned the firearm, portrayed by observers as a siphon activity shotgun, on himself.
Preceding the shooting, Davison had posted disdain filled online tirades about single parents and about his own mom specifically, calling her “awful, useless and tumultuous”.
The IOPC said it had dispatched an examination on Friday evening after it got primer data that Davison’s gun and permit were gotten back to him toward the beginning of July this year.
The declaration and shotgun had been eliminated from him by police in December 2020 after a claim of attack in September 2020, the guard dog said.
Executive Boris Johnson had before said the issue of how Davison came to lawfully possess a firearm ought to be “appropriately explored”, adding that the shooting was an “totally shocking” episode.
Territorial IOPC chief David Ford said: “We will analyze what police moves were made and when, the reasoning behind police dynamic, and regardless of whether pertinent law, strategy and systems were followed concerning Mr Davison’s ownership of a shotgun.”
It will likewise take a gander at whether the power had data concerning Davison’s psychological wellness and in case this was “suitably thought of”.
Boss Constable Shaun Sawyer revealed to Sky News that those present had confronted “the absolute most testing scenes”.
“We’ve never in my time had murder followed by a rampaging guns assault on irregular individuals from general society and afterward ending one’s life,” Mr Sawyer said.
“That is unprecedented in my time as boss constable.”
Police are not treating the episode, the most exceedingly awful mass shooting in Britain since 2010, as dread related.
Notwithstanding, Davison made references to “incels” in some online web-based media recordings.
Incels are individuals from misanthropic online gatherings of “compulsory abstinent” men, who fault ladies for their sexual failings and who have been connected to various savage demonstrations all throughout the planet.
Officials are proceeding to research 13 scenes inside a cordon that is relied upon to stay set up all through a large part of the end of the week.
On Friday evening, many individuals set blossoms and candles in North Down Crescent Park.
One note read: “Find happiness in the hereafter. Musings and supplications with all of you. A heartbreaking misfortune. Such excellent spirits. Love to all.”
Plymouth Sutton and Devonport MP Luke Pollard tweeted pictures of the vigil and the casualties’ names, expressing: “This evening our local area met up to recollect the survivors of the shooting in Keyham.”
Banners in Plymouth are being flown at half-pole, while Smeaton’s Tower was illuminated as a characteristic of regard to the casualties on Friday night.