Selectors promoted wicketkeeper-batsman Inglis ahead of senior players including Glenn Maxwell, Adam Zampa and Marcus Stoinis, Reuters reported.
ODI skipper Cummins will be rested for the third and final one-dayer in Perth on Sunday along with test regulars Mitchell Starc, Josh Hazlewood, Marnus Labuschagne and Steve Smith.
T20 captain Marsh is on paternity leave.
"Josh is an integral member of the ODI and T20I teams and a highly respected player on and off the field," selector George Bailey said in a statement on Wednesday.
"He has led Australia A previously and will bring strong tactical nous and a positive approach to the role."
None of Australia’s test players will feature in the three-match T20 series, which starts on Nov. 14 at the Gabba in Brisbane, as they warm up for the five-test series against India beginning on Nov. 22 in Perth.
Australia took a 1-0 lead in the three-match ODI series with a two-wicket win over Mohammad Rizwan’s Pakistan at the Melbourne Cricket Ground on Monday.
The hosts will look to seal the series with victory in the second match at Adelaide Oval on Friday.
Selectors have recalled pace stalwart Hazlewood for the second ODI after he missed the MCG match to play Sheffield Shield cricket.
Pace bowlers Xavier Bartlett and Spencer Johnson, and wicketkeeper-batsman Josh Philippe, have been added to the squad for the third ODI.
Australia have yet to decide on a test opener to replace David Warner, who has retired from international cricket, but Bailey hinted at the leading candidates in revealing the batting order for the second Australia A match against India A.
Australia A captain Nathan McSweeney has been promoted to open with 14-test batter Marcus Harris for the four-day match starting at the MCG on Thursday, having scored 39 and an unbeaten 88 in the first fixture at Mackay.
Young gun Sam Konstas has been dropped down to number four after managing a total of 16 runs in Mackay, while former test opener Cameron Bancroft remains at number three after exactly matching Konstas’s output with the bat.
While not a specialist opener, South Australia captain McSweeney has also been in good form early in the Sheffield Shield, scoring a century and two fifties in two matches of the domestic first class competition.
India, meanwhile, are sending batters KL Rahul and Dhruv Jurel to Australia early so they can warm up for India A before the test series, Indian media reported.
Rohit Sharma’s India were whitewashed 3-0 by New Zealand, the first time they have been swept in a three-match test series on home soil.
