To help fund its operations, the Sierra Club may start a mutual fund to invest in companies that pass a strict green test. The group’s executive director, Carl Pope, said the fund would use a tougher investment screen than most other mutual funds marketed as being environmentally responsible. The fund would be managed by an outside firm and pay royalties to the club for the use of its name and investment screen. The club hopes to earn as much as $1 million from the fund within five years. But some board members haven’t yet bought into the idea. “Selling the club’s name cheapens it,” said Phillip Berry, a former president, at a board meeting in May.