The police watchdog has said the Metropolitan Police's professional standards unit should investigate the force's handling of a case after the death of a man suspected of being the "Putney Pusher".
Nicholas Brandram was found dead at his London home on 15 September. His family said he took his own life after the "immense pressure" of a police investigation.
The 44-year-old was under suspicion of being the jogger who pushed a woman into the path of a bus during a run on Putney Bridge in west London in May 2017.
The Met police made a mandatory referral to the Independent Office for Police Conduct (IOPC) following the banker's death.
The family of Nicholas Brandram said a Metropolitan Police investigation into its handling of the case should be "thorough, independent and unflinching in establishing what went wrong".
His sisters Alexia Hicks and Sophie Voelcker said in a statement: "We have been clear about the litany of failings that need to be examined, and we expect those questions to be answered."
They called for the investigation to be a "priority", adding that they wanted no other family "to endure what ours is going through".
In a statement, the IOPC said: "After carefully considering the referral, we have decided that an investigation is required and that the Met's Directorate of Professional Standards should conduct it.
"We have recommended the review considers matters including the necessity and proportionality of arrest, what engagement was there with the media and considerations made in relation to Mr Brandram's welfare and the relevant policy."
Brandram had been arrested in June on suspicion of attempted grievous bodily harm in relation to the incident in May 2017.
He was linked to the incident in April this year when a person contacted officers with information that opened new lines of inquiry, according to the Met.
His family has since claimed there is no forensic evidence linking him to the case.
The Met said that it had not named Brandram publicly when he was arrested or at any point before his death.
Ch Sup Dan Knowles said he recognised the "strain that being subject to a police investigation" could cause to individuals and their families.
He confirmed that the Met's Directorate of Professional Standards would carry out the investigation.
Video footage from May 2017 appeared to show a jogger pushing a woman into the road as a bus approached. The woman narrowly avoided being hit.
Bus driver Oliver Salbris told the Sunday Times at the time: "If I hadn't swerved, I would have smashed her head."
Detectives identified more than 50 people of interest following the incident.
The Met had initially closed its investigation in 2018 after three men had been arrested and released without charge.
The force said the investigation into the incident remained ongoing.
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