Overnight co-leader Paul Dunne of Ireland opened a cushion over the field with a Friday 65, his nine birdie-two bogey effort taking him to 13-under for the week.
Rumford added a 66 to his opening 68 to reach 10-under and be alongside Spain’s Nacho Elvira and England’s Callum Shinkwin.
Tournament drawcard and local favourite Jon Rahm thrilled the crowd with some final hole heroics, hitting the flag with his second at the par-5 18th on his way to an adventurous 68.
The World Number Four opened with an eagle at the first hole and added two birdies before a clumsy double bogey at the 12th stopped his momentum.
He got both shots back by the end of the round but at four behind may rue an aggressive approach at the par-4 which found water beyond the green.
Jon Rahm finishes in style! pic.twitter.com/wTri9IdW1p— The European Tour (@EuropeanTour) April 13, 2018
Rumford is one of three Australians in the top 20 on a bunched leaderboard, Marcus Fraser matching Rumford’s 66 to be 7-under and T14 while Jason Norris returned a gritty 71 to be 6-under and T20.
Jason Scrivener is the last of the seven Australians in the field to make the cut, his 5-under total one better than required to play the weekend.
New Zealand’s Ryan Fox and Japan Tour regular Adam Bland both missed by two shots, Fox making two late bogeys to miss out while Bland made double at his penultimate hole.
Nick Cullen and Andrew Dodt both started Friday well back and slid further behind with rounds of 72 and 73 respectively.
Meanwhile for Dunne, the three shot lead represents the first time in his short career he has been atop the leaderboard through 36 holes.
“I wasn't really thinking about the score, then the score kind of built on itself,” he said.
"The conditions were a lot better than the last couple of days and playing early I got the best of the greens so it was a pretty steady day.
“My game is in good shape, I'm feeling good. I see a lot of birdies around this course if the weather stays like this so hopefully I can make them.”
England’s Andy Sullivan produced the best score of the day with a 9-under 63 to make up for an opening 75 to make the cut by two shots.
WHAT YOU CAN LEARN FROM RICKIE FOWLER'S SWING:
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