Rumford had six birdies over his first 10 holes in Friday's second round to get to a 9-under 135 for the tournament. Bradley Dredge, who shared the first-round lead, had a 3-under 69 Friday and is alone in second one stroke behind Rumford.
Oliver Wilson, the other first-round co-leader, is tied with Lee Westwood for third at 7-under 137.
Rumford, playing the Doha Country Club's back nine first Friday, opened with a birdie at No. 10 and added another three holes later. He got to 3-under on the day with a birdie at 15 before running off consecutive birdies in 17, 18 and the first hole.
He dropped a stroke at the par-4 fifth hole but ended the round with his seventh birdie of the day.
The birdie at 18 had come on a chip-in and at No. 9 Rumford had a chip shot that set up a short tap in to end the round with a birdie.