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The purpose of this paper is to study minimal modular extensions of braidedfusion categories doing emphases in minimal modular extensions ofsuper-Tannakian fusion categories. We define actions of finite super-groups onfermionic fusion categories and spin-braided fusion categories. Similar to thecase of groups, our motivation came from the study of fusion categoriescontaining the representation category of a super-group. We develop many analogresults to the Tannakian case, including cohomological obstructions, relationwith braided $G$-crossed categories and minimal modular extensions. We applythe general results to the construction and classification of minimal modularextensions of super-groups and braided fusion categories. In particular, weexhibit some examples of braided fusion categories without minimal modularextensions.

Categorical Fermionic actions and minimal modular extensions