[Fascism has an extremely hard time rising under authoritarianism]

Maybe any non-authoritarian control does ALWAYS result in fascism.

You're misunderstanding the terminology. Liberalism, as it's being used here, is synonymous with capitalism. It's not the opposite of authoritarianism.

In the historical examples of Germany and Italy, fascism came about out of a capitalist system and supported and was supported by the capitalist class. The economic policies of fascism involved nationalizing businesses belonging to the out groups, but was primarily focused on privatization, including the aforementioned businesses, in order to redistribute them to the in group. The reason that the capitalists supported fascists was because the systems they were in had become unstable and unsustainable, and they knew that their interests would be threatened if socialists came to power, but not if fascists did.

That's not to say that every capitalist system always leads to fascism, but rather that every fascist state has come out of capitalism. Of course, people have different definitions for fascism, and the degree that that's true depends on how you define it. A narrow definition that treats a capitalist class as a defining characteristic of fascism would make it true by definition.