Our first objective is to present equivalent conditions for the solvability of the system of matrix equations ADA = A, DAB = B and CAD = C, where D is unknown, A, B, C are of appropriate dimensions, and to obtain its general solution in terms of appropriate inner inverses. Our leading idea is to find characterizations and representations of a subclass of inner inverses that satisfy some properties of outer inverses. A G-(B, C) inverse of A is defined as a solution of this matrix system. In this way, G-(B, C) inverses are defined and investigated as an extension of G-outer inverses. One-sided versions of G-(B, C) inverse are introduced as weaker kinds of G-(B, C) inverses and generalizations of one-sided versions of G-outer inverse. Applying the G-(B, C) inverse and its one-sided versions, we propose three new partial orders on the set of complex matrices. These new partial orders extend the concepts of G-outer (T, S)-partial order and one-sided G-outer (T, S)-partial orders.