On F-inverse covers of inverse monoids

It is shown that each finite inverse monoid admits a finite F  -inverse cover if and only if the same is true for each finite combinatorial strict inverse semigroup with an identity adjoined if and only if the same is true for the Margolis–Meakin expansion M(H)M(H) of each finite elementary abelian p-group H for some prime p. Additional equivalent conditions are given in terms of the existence of locally finite varieties of groups having certain properties. Ultimately, the problem of whether each finite inverse monoid admits a finite F  -inverse cover, is reduced to a question concerning the Kostrikin–Zelmanov varieties KnKn of all locally finite groups of exponent dividing n.