Labor's shadow attorney-general Mark Dreyfus wrote to the NSW Police last month asking for an investigation into the matter,arguing that using a forgery to influence an elected representative was a breach of the NSW Crimes Act.
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The formation of Strike Force Garrad was revealed on Tuesday.
Federal Opposition Leader Anthony Albanese has called on Mr Morrison to require Mr Taylor to stand aside while the investigation is under way,citing the ministerial code of conduct updated in August 2018.
Clause 7.1 of the code says"ministers must accept that it is for the prime minister to decide whether and when a minister should stand aside"if the minister becomes the subject of an official investigation of alleged illegal or improper conduct.
Clause 7.2 says ministers will be required to stand aside if charged with any criminal offence,or if the prime minister regards their conduct as constituting a prima facie breach of the standards. They are required to resign if convicted of a criminal offence.
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Mr Morrison stood by Mr Taylor on Tuesday after telling the House of Representatives he had phoned the NSW Police Commissioner after earlier telling Parliament he would check on the investigation.
"I take matters of ministerial standards very seriously,"Mr Morrison said.
"I have since spoken with the NSW Police Commissioner,Mick Fuller,about the instigation,the nature and the substance of their inquiries,which he advised me were based only on the allegations referred by the shadow attorney-general.
"Based on the information provided to me by the Commissioner,I consider there is no action required by me under clauses 7.1 and 7.2. The NSW Police should now be left to complete their inquiries,which will be considered upon their completion."
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Mr Fuller told radio station 2GB last year of his friendship with Mr Morrison when they were neighbours in Sydney.
Centre Alliance senator Rex Patrick said Mr Morrison's phone call was"even more inappropriate"because of that friendship and that checks on the status of Strike Force Garrad could have been made through Home Affairs Minister Peter Dutton.
"A staffer could have made that call,"Senator Patrick told Sky News.
"A staffer could have gotten in contact,even through formal channels in Mr Dutton's office,perhaps through the AFP. It didn't have to be the Prime Minister,"Senator Patrick said.
"Because they know each other,they have to be very,very careful. It creates all sorts of perceptions,and,as we know in politics,it's not only about being clean,it's also about being seen to be clean."
The dispute over the phone call triggered a response from Mr Fuller on Wednesday as the government prepared for another Labor attack in question time.
Mr Fuller conducted an interview withThe Australian newspaper in which he said it was wrong to claim the phone call was inappropriate.
"You can't have a police commissioner making decisions to[conduct] an open and transparent investigation,then suggest there's something inappropriate in relation to receiving a phone call. It's hypocrisy to say that,"Mr Fuller said.
"It was an extremely short conversation in that he just wanted confirmation we were conducting an investigation."