Categorical Fermionic Actions and Minimal Modular Extensions

We define fermionic actions of finite super-groups on fermionic fusion categories and establish necessary and sufficient conditions for their existence. Our main result characterizes when a braided fusion category admits a minimal non-degenerate extension in terms of cohomological obstructions. This characterization for braided fusion categories with non-Tannakian M"uger center involves the fermionic structures and fermionic actions introduced in this work.