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Let R be an arbitrary ring with identity and M a right R-module with S = EndR(M). Let F be a fully invariant submodule of M. We call M an F-inverse split module if f−1(F) is a direct summand of M for every f ∈ S. This work is devoted to investigation of various properties and characterizations of an F-inverse split module M and to show, among others, the following results: (1) the module M is F-inverse split if and only if M = F ⊕ K where K is a Rickart module; (2) for every free R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split if and only if for every projective R-module M, there exists a fully invariant submodule F of M such that M is F-inverse split; and (3) Every R-module M is Z2(M)-inverse split and Z2(M) is projective if and only if R is semisimple.