Rumford, a native of Perth, rolled in eight birdies, including five straight from the sixth hole, and a lone bogey to finish his round at a seven-under-par 65. That put him at 13 under for the tournament after his opening-round 66 at the Lake Karrinyup Country Club.
David Bransdon fired a 64 to come in at 11 under, a stroke ahead of Steven Jeffress at third.
The overnight joint leader, England's Mark Foster, had a round of one over on Friday that left him tied for 26th place. World number 11 Alex Noren, the highest-ranked player in the field, missed the cut after a two-day score of one under.
British Open winner Louis Oosthuizen stayed in the hunt with a second-round 68, which left him tied for 10th at seven under.
A total of 67 players, including one amateur, made the regular cut in the event, tri-sanctioned by the PGA Tour of Australasia, Asian Tour and European Tour.
After the usual three days of stroke play, the A$1.75 million (£1 million) tournament will end on Sunday with 24 players competing at match play in six-hole playoffs until only one remains.
The top eight after tomorrow's third round will earn a seeding into the fourth round and get a bye during the first round of match play. Rumford is keen to be one of those eight.
"That's one less lottery round you have to deal with, isn't it?" Rumford said. "That's definitely a huge bonus qualifying inside that top eight, so it's definitely something I am thinking about.
"That's why it's going to be so unique and so interesting as well. There's sort of cuts within cuts this week."
(Corrects Rumford score on Friday to 65, not 63.)
(Reporting by Sudipto Ganguly; editing by Larry King)
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