Kumar, 24, carded a six-under-par 66, one off the course record, to finish on 11-under 133.
Briton Simon Hurd was in second place after compiling a 70 for 137. South Africa's Simon Griffiths (72) and Indian Firoz Ali (73) were joint third, a further three strokes adrift.
South African Hendrik Buhrmann, the first-round leader, carded five bogeys in a three-over 75 to slip to joint fifth with Australians Marcus Both and Jarrod Moseley and American Bryan Saltus.
Kumar almost aced his first hole, the par-three 10th, when his eight-iron rolled to within a foot off the pin to earn the first of his seven birdies at the picturesque course near Mumbai.
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Hurd's game fluctuated for the second consecutive day as his five birdies were offset by three dropped shots.
"It's windier in the afternoon so the course plays a bit tougher and the fairways play a bit tighter," he said. "Good luck to the Indian guy who is leading, I'll try to reel him in."
A total of 67 players made the cut at four-over in the inaugural tournament, one of two Asian Tour events in India.
