A gunman who killed his mother and four passersby, including a three-year-old girl, had his firearms licence revoked in December, but police reinstated it last month after he attended an anger management course.

Police will face an investigation over their dealings with Jake Davison, 22, who expressed sympathy for the “incel” movement and a keen interest in mass shootings. One resident from Plymouth, where the killing spree took place, said Davison’s family had sought treatment for his mental health issues.

Detectives are also facing questions over the decision to treat the attack – which was Britain’s worst mass shooting in over a decade – as a domestic incident rather than a terror attack. On Friday evening sources said the main motives under investigation were a hatred of women and mental heath issues.

As well as his mother, Maxine, 51, Davison’s victims were named on Friday as Lee Martyn, 43, and his three-year-old daughter Sophie Martyn, whom he is believed to have adopted, and Stephen Washington, 59, and Kate Shepherd, 66.

Devon and Cornwall police suspended Davison’s firearms licence in December last year over concerns about his suitability to hold a weapon, but deemed him suitable to have his gun back last month. He had held a licence since 2018.

The Guardian understands Davison had to attend an anger management course to get his licence back, and months ago was classed by police as being fit again to possess the three-shot shotgun, despite earlier concerns.

In 2020 he had a row with two youths and was reported to police, but he had no previous offences and agreed to attend the course, sources said. The police watchdog, the Independent Office for Police Conduct (IOPC), will formally investigate the handling of his gun licence by police.

Davison had shared hate-filled views on Reddit forums used by incels – men who express hostility and resentment towards those who are sexually active, particularly women. Earlier this year, authorities in the US warned that attacks linked to the incel movement were on the rise, and authorities around the world have begun to treat the ideology as a more serious terrorism threat.

Jonathan Hall QC, the independent reviewer of terrorism legislation, told the Guardian: “Incel(dom) definitely can be an ideology for the purposes of terrorism. It always depends on the evidence, but if you kill people in a symbolic way because you hate what they represent within your ideology, it’s not a big stretch to conclude that killing them is in pursuance of an ideological cause.”

Counter-terrorism policing declined to comment and referred inquiries to Devon and Cornwall police, which said they were keeping an open mind and that there was “no [known] motive” at present.

Davison, an apprentice at a defence and security company, turned the gun on himself after killing five people in 12 minutes soon after 6pm on Thursday. He is thought to have used a pump-action shotgun that he reloaded at least twice during his rampage and for which he had had a firearms licence for three years for sports use.

He is understood to have been known to police in his youth, though not for anything serious enough to permanently ban him from owning a gun.

At a press conference in Plymouth, Shaun Sawyer, the chief constable of Devon and Cornwall police, said Davison murdered his mother at her home in Biddick Drive, which he is thought to have shared with her, before leaving the property and shooting dead the Martyns in the street.

Davison then shot two other people, a man and woman aged 33 and 53 respectively, in Biddick Drive. They are in hospital with significant but not life-threatening injuries.

He entered adjacent parkland where he shot dead Washington. Then, in Henderson Place, he shot Shepherd, who later died at Derriford hospital in Plymouth. He killed himself as police closed in.

Davison had written about guns online, including contributing to a Reddit chat called “GunPorn”. In one post he said he craved owning a VZ58 assault pistol. In another he said his weapon of choice would be a Glock 17.

Davison used incel forums to express hatred for his mother and a view that mass shootings had no connection to gun control. In YouTube videos, Davison said he did not “clarify” himself as an incel – but went on to talk about how “people similar to me have had nothing but themselves”.

He appears to have signed up to “IncelTV” on YouTube, which calls itself a channel for “black pill and lookism content” and has more than 17,000 subscribers. Lookism is described as prejudice or discrimination on the grounds of a person’s appearance. In incel culture, someone who has “taken the black pill” believes they are a hopeless case who will never have romantic or sexual success.

Social media posts from about 2018 suggest Davison was a fan of Donald Trump and a supporter of the UK Libertarian party. YouTube and Facebook confirmed that accounts belonging to Davison were taken down on Friday, and people were asked not to share footage of the aftermath of the Plymouth attack.

The condition of Davison’s mental health has been questioned, with one person familiar with his family claiming relatives had requested help from mental health authorities. Writing on Facebook, the neighbour said: “His family [pleaded] for help to the mental health team, the NHS basically said that they are short-staffed and that was it. The family even asked for the police to come out to see him as he was talking and acting strange but they didn’t do a welfare check.”

The Guardian approached community mental health services in Plymouth but they declined to comment.

A spokesperson for Babcock International, a defence and security company that has a significant presence at the city’s dockyards in Devonport, confirmed that Davison had worked for the firm as an apprentice since August 2020. The company’s CEO, David Lockwood, earlier said: “Our thoughts and condolences are with the friends and families of those involved in this tragedy. We stand with the city at this difficult time.”

The IOPC said it had received a mandatory referral from Devon and Cornwall police relating to the incident. The IOPC regional director David Ford said: “After assessment of the referral we have determined we will carry out an independent investigation focusing on Jake Davison’s firearms licensing history and its impact on the tragic events of Thursday 12 August.

“We will examine what police actions were taken and when, the rationale behind police decision-making, and whether relevant law, policy and procedures were followed concerning Mr Davison’s possession of a shotgun. The investigation will also consider whether the force had any information concerning Mr Davison’s mental health and if so, if this information was appropriately considered.”

Tributes were paid to the friends and family of the deceased. Boris Johnson tweeted: “My thoughts are with the friends and family of those who lost their lives and with all those affected by the tragic incident in Plymouth last night. I thank the emergency services for their response.”

Labour’s Keir Starmer tweeted: “My thoughts are with the families and neighbours of those caught up in this nightmare. I pay tribute to our emergency services who ran towards events we’d all run from.”