Parity-conserving and -violating response functions are computed for the inclusive electroexcitation of the N * (1440) (Roper) resonance in nuclear matter modeled as a relativistic Fermi gas. Using various empirical parameterizations and theoretical models of the N-N * (1440) transition form factors, the sensitivity of the response functions to details of the structure of the Roper resonance is investigated. The possibility of disentangling this resonance from the contribution of Δ electroproduction in nuclei is addressed. Finally, the contributions of the Roper resonance to the longitudinal scaling function and to the Coulomb sum rule are also explored.