Australian Prime Minister Anthony Albanese has called on authorities in Pakistan to investigate the fatal shooting of an Australian girl while on holiday with her family.

Nine-year-old Hania Ahmed and her family were taken hostage at gunpoint by armed robbers inside their rental car in the northern city of Chakwal in Punjab province on 10 June.

The suspects fired at a police officer, leading to a shootout, Punjab police said.

A police officer, thinking that the suspects were fleeing on the family's vehicle, "mistakenly" fired his gun, resulting in the nine-year-old's death and wounding her father and older brother, police said.

Punjab police said the suspects fired first at the police officer, who has been placed under custody. However, the girl's father told SBS Urdu that the police fired first.

"These circumstances do need to be examined. They need to be examined in a transparent way, so that everyone can know, the family, most importantly, but others as well," Albanese told reporters in Canberra on Monday.

"Australia expects there to be transparency and a proper investigation of these circumstances," he said.

Punjab police said there was "absolutely no justification for deviating from our established protocols" and that it was conducting a "thorough and impartial investigation".

The girl's death has shocked both Australia and Pakistan.

Abdullah Khan, the principal of Hania's school in Perth, the Australian Islamic College, told the BBC that news of her death was "traumatic" for her community.

Hania was very friendly, bubbly, and very social, Khan said.

"She had lots of friends, [and was] very respectful to teachers," he added. "She was loved by everyone."

Khan said he has been in touch with Hania's family, and that the school was providing counselling support to staff and students.

"Especially the students in her class - they are distressed and in a state of shock," he said.

The Australian goverment is providing consular assistance to the family.

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