Matt Hendrix carded a 5-under 67 to climb into third place at minus-6. Steven Alker had the low round of the day as he shot a 7-under 65. That moved him to 5-under-par 139 where he shares fourth place with Chris Tidland, who shot 69 in round two.
The first and second rounds were both completed on Friday. The first round was suspended due to darkness Thursday after there was a two-hour weather delay early in the round at Clearwater Resort.
Lyle, who played the final four holes of his first round earlier in the day, played the back-nine first Friday and came out on fire. He birdied three straight from the 11th to quickly move to four-under par for the championship.
The Australian picked up the fourth birdie of his round at the par-four 15th. However, Lyle faltered to a bogey at the 18th to make the turn at four-under. He got that stroke back with a birdie on the second.
Lyle made it two straight to move to six-under. He notched another birdie at the sixth. Lyle bogeyed the eighth, but closed with a 35-foot birdie putt on the ninth to grab a share of the lead.
'I had a good finish last week and holed a lot of putts there,' said Lyle, who tied for fourth last week at the Jacob's Creek Open Championship. 'The putter felt a bit average Thursday, I didn't hole that many putts. But today, something sort of clicked. I just did my normal routine and just putted real solid today. The putter's feeling really good.'
Rumford, who parred the final two holes of his first round earlier Friday, also started on the second nine and parred his first four holes. He birdied the par-five 14th to get to four-under. He parred the remaining four holes of his opening nine.
On the front nine, Rumford birdied the par-five second. He got to six-under with a birdie at the par-three fourth. The Australian picked up his final birdie at the eighth to end at minus-seven.
'It was a good day. I obviously continued the same form from Thursday,' said Rumford. 'I've been driving it well, which I think is key around here with the rough up. All in all it was a great day.'
Kevin Stadler, who won the Johnnie Walker Classic on the European Tour two weeks ago, shot 69 in round two and stands alone in sixth place at four-under- par 140.
First-round leader Jason Dufner struggled to a two-over 74 Friday. He slipped in to a tie for seventh at minus-three. He was joined there by Aron Price, Joel Edwards, Johnson Wagner, Scott Weatherly, Keoke Cotner, Nick Flanagan, Paul Gow and Stephen Allan.
The cut line fell at two-over-par 146 with 70 players advancing to the weekend. Among those who missed out on the final two rounds was 2005 champion Peter O'Malley, who was plus-five over the first two days.