BENGALURU, April 17 (Reuters) - Investment firms GQG and Fidelity picked up shares worth $261 million in India's Vodafone Idea as part of the debt-saddled telecom operator's planned $2.16 billion share sale, the company said.

Vodafone Idea sold shares worth 54 billion rupees ($645.5 million) to institutional investors in the first stage of its 180 billion rupees follow-on offering, the country's largest ever, to raise funds to expand it services.

U.S.-based GQG, run by India-born executive Rajiv Jain, was allocated shares worth 13.48 billion rupees, the most among all institutional investors, according to Vodafone Idea's statement late on Tuesday.

Fidelity was allocated shares worth 8.32 billion rupees, while Motilal Oswal got shares worth 454.5 million rupees, the most among domestic investors, the statement showed.

The shares were sold at 11 rupees each, the upper end of the 10-11 rupees price band. Vodafone Idea's stock ended at 12.95 rupees on Tuesday, the day of the bidding.

The bidding for retail investors will be at the same price band and run from Thursday, April 18 until April 22. Indian markets are closed on Wednesday for a public holiday.

Vodafone Idea has said it plans to raise 450 billion rupees, via equity and debt, to help expand its 4G network and roll out 5G services.

Reuters reported last week that GQG was planning to invest in Vodafone Idea, India's third-biggest operator after Reliance Jio and Bharti Airtel, which have taken away its market share in recent years.

GQG's Jain has a recent track record of backing troubled companies with falling share prices and profiting from them. His recent successful bet was on Indian conglomerate Adani Group.

GQG manages more than $100 billion globally and $20 billion in India, with Jain striking a bullish note on the country's economic prospects. ($1 = 83.6590 Indian rupees) (Reporting by Nandan Mandayam in Bengaluru; Editing by Savio D'Souza)