Lifting modules as a main concept in module theory have been studied and investigated extensively in recent decades. The rst author in [1] tried to consider and investigate this concept with a homological approach. Let R be a ring and M be a right R-module. Then M is called I-lifting if image of every endomorphism of M lies above a direct summand of M. In this paper, we are interested in studying modules M with the property that '(F)=D M=D for every endomorphism ' of M and for some direct summands D of M, where F is a xed fully invariant submodule of M. We call such modules IF -lifting. We provide some examples of IF -lifting modules as a proper generalization of lifting modules. Some characterizations of IF -lifting modules are given. We also de ne relative IF -lifting modules to study direct summands and nite direct sums of IF -lifting modules.
