We study optical Kerr frequency combs in nonlinear microresonators in the presence of two-photon absorption (TPA) and free-carrier absorption (FCA) described by an extended Lugiato-Lefever system of equations. We give a criterion for modulation instability of constant stationary solutions and establish existence of non-constant solutions. Both results are accompanied by numerical simulations, which show that the abstract conditions both for modulation instability and existence are satisfied in large parts of the parameter regions.