  Motivated by weak ferromagnetism (FM) in a $\tau$-type molecular conductor
($\tau$-MC), we examine its mechanism using a two-band extended Hubbard model.
Applying the random phase approximation, we elucidate the uniform spin and
charge fluctuations between unit cells in the presence of on-site and off-site
interactions. Applying the mean-field approximation, we find the ordered state
mixing with antiferromagnetism (AFM), weak FM, and charge ordering (CO)
components in each unit cell: we classify this state as ferrimagnetism (FIM).
We reveal the phase diagrams in the interaction and interaction-temperature
spaces. The former shows that the off-site interaction induces FIM from pure
AFM and the latter shows that lowering the temperature stabilizes FIM. To
clarify the stabilization mechanism of the phases, we focus on the microscopic
nature of the ordered states, including the band structure, Fermi surface, and
density of states. We find that the FIM state is obtained from mixing features
of AFM, CO, and FM; therefore, the emergence of FIM requires both the on-site
and off-site interactions. Then, we discuss the effect of lowering the
temperature and predict that the AFM gap assists the emergence of FIM based on
AFM. This FIM state is possibly related to the observation of the weak FM in
$\tau$-MC.
