We define two categories, the category FG of fuzzy subgroups , and the category FC of F-inverse covers of inverse monoids, and prove that there is a full and faithful embedding of FG into FC. As a by-product of this embedding we get that the level subgroups of a given fuzzy subgroup can be realized as the H-classes of a Clifford monoid that is canonically constructed from the fuzzy subgroup. This connection we find between fuzzy subgroups and inverse monoids is new and unexplored before and shows that, at least from a categorical viewpoint, fuzzy subgroups belong to the standard mathematics as much as they do to the fuzzy one.