Kapur, one over for the first round, is now one-under for two rounds and five shots behind the leader Marcel Siem, who carded successive 68s in the first two rounds.
Also making the cut was Jeev Milkha Singh despite a rough three-over 74 after a first round three-under 68. At even par he is now tied 26th.
However, SSP Chowrasia continued to struggle on European soil with a nightmarish 83 that saw him at 15-over for two rounds and crash out of the event.
Meanwhile, Marcel Siem of Germany will take a one shot lead into the third round with a late surge by Alexander Noren at Le Golf National.
Siem, sixth on home soil at last month's BMW International Open, went ahead courtesy of a second successive 68 on a day when play was suspended for around 80 minutes.
Siem is now six-under for two rounds.
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Kapur after a double bogey on the first hole recovered with back-to-back birdies on fifth and sixth. On the back nine he had two more birdies on 12th and 14th.
Jeev had four bogeys in his first 10 holes and had to wait till the 16th hole for his first and only birdie of the day on a rough day.
Siem was briefly joined by Noren after the Swede recorded birdies at the 14th, 15th and 16th to reach five under for his second round.
But Noren found water at the last and made bogey to drop back into second, one clear of Denmark's Anders Hansen and South African George Coetzee.
David Lynn, Francesco Molinari and Raphael Jacquelin shared fifth on three under.