BENGALURU, Dec 29 (Reuters) - Indian shares extended a foreign inflows-powered rally on Tuesday, with beaten-down banking stocks gaining the most on the back of an improving economic outlook, although a fall in Reliance Industries capped gains. Banking, which is among the few sectors still down this year, has been catching up with the broader market’s advance as foreign investors bet that a rebound in economic activity will benefit lenders. The Nifty Bank index rose 1.23%, with HDFC Bank Ltd, the nation’s largest private-sector lender, gaining 1.3%. The stock was on track to finish higher for a fifth straight session. “Large private-sector lenders have been proving themselves time and again with good asset quality and decent growth,” said Neeraj Dewan, director at New Delhi-based Quantum Securities. “When foreign institutional investors come to buy, these lenders are normally on top of their list,” he said. India’s NSE Nifty 50 index was up 0.07% at 13,884.70 by 0530 GMT, while the S&P BSE Sensex was 0.12% higher at 47,403.81. Both indexes hit record intraday highs before pulling back by midday trade as Reliance Industries Ltd, India’s most valuable company, ended a three-session rally, falling 0.8%. Metal and energy stocks lost ground, with the Nifty Energy index and the Nifty Metal index slipping about 1% each. Hopes that a U.S. pandemic relief package would be expanded were aiding investor sentiment globally, boosting broader Asian shares. Separately, India reported six cases of the new, more infectious variant of the coronavirus, all from people who had returned from Britain. However, the daily rise in overall infections in India has been slowing. (Reporting by Chandini Monnappa in Bengaluru; Editing by Vinay Dwivedi)