The network, which helps people set up schools outside local authority control, was granted charitable status in October last year.
It was founded by Rachel Wolf, a former political adviser to Michael Gove, the Secretary of State for Education, who wants to establish more academies and ‘free’ schools.
Gove’s department awarded the network £500,000 in June to advise groups that are interested in setting up free schools.
Martin Johnson, deputy general secretary of the ATL, told Third Sector the network was in effect "implementing government policy".
He asked: "Should a body that does this be allowed to describe itself as a charity and enjoy the benefits this brings? The relationship between this organisation and the policy is quite blatant."
Wolf said the charity was funded in an identical way to many other charities.
"A lot of charities support particular policies and even campaign for these policies," said Wolf. "That doesn’t make them party political. "We haven’t done anything that’s against charity law."
On her links with Gove, she said: "I don’t think you can ban people from working for a charity because of their previous jobs."
A commission spokeswoman said it had not received any complaints about the network.