HESTA, the industry super fund for people in health and community services with more than one million members, has since 2022 had both Woodside and Santos on its watchlist, meaning they are subject to increased engagement and monitoring.
Market Forces, Healthy Futures, Doctors for the Environment Australia and the Climate and Health Alliance say that's not enough given that the pair, Australia's two biggest oil and gas developers, are moving ahead with new fossil fuel projects.
"It is pretty hypocritical for a health sector super fund to in any way support the fossil fuel industry," said Kate Wylie, a general practitioner and executive director of Doctors for the Environment Australia.
"Coal oil and gas are causing immense harm to the health of our communities and our environment, so it's a clear breach of our duty of care as doctors to have our super invested in them.
"HESTA does not invest in tobacco or nuclear weapons, so why invest in fossil fuels?""
HESTA's general manager of responsible investment, Kim Farrant, said that the super fund, which has close to $98 billion in assets, directly engages with company leadership to push for greater climate action and transparency.
"We strongly believe this active ownership approach is more effective than divestment alone at driving change,'' she said.
"Selling shares without attempting to change company behaviour simply pushes responsibility to the next buyer, who may not share our view that greater action is required on climate change."
HESTA has delivered significant outcomes through this approach, including the uptake of net zero targets, disclosure of climate-related risk and initial investments in breakthrough technologies, Ms Farrant said.
Brent Morgan, the superannuation funds campaign lead for Market Forces, said it's been three years since HESTA asked Santos and Woodside to demonstrate how their business plans align with global climate goals.
"Since then they haven't taken any effective action to reign in these companies' oil and gas expansion plans," Mr Morgan told AAP.
Like a number of other superannuation funds, HESTA has cast votes against what Woodside and Santos's board of directors have recommended in recent years.
This year, it cast votes against Woodside's executive pay plan and against the re-election of company director Ann Pickard, who did win re-election despite nearly 20 per cent opposition.
"While HESTA has been outspoken in the past on issues including climate change, at this stage, that's all it appears to be - we're not seeing any effective action to bridge that gap between what HESTA says what it ought to be doing," Mr Morgan said.
Ms Farrant said that HESTA was supporting change in the real economy, including advocating for an ambitious 2035 emission reduction target for Australia.
"We believe our strong commitment to responsible investing is in members' best financial interests and has supported the delivery of strong long-term returns for our members," she said.