  We develop a model of relativistic, charged meson-exchange currents (MEC) for
neutrino-nucleus interactions. The two-body current is the sum of seagull,
pion-in-flight, pion-pole and $\Delta$-pole operators. These operators are
obtained from the weak pion-production amplitudes for the nucleon derived in
the non-linear $\sigma$-model together with weak excitation of the
$\Delta(1232)$ resonance and its subsequent decay into $N\pi$. With these
currents we compute the five 2p-2h response functions contributing to
$(\nu_l,l^-)$ and $(\overline{\nu}_l,l^+)$ reactions in the relativistic Fermi
gas model. The total current is the sum of vector and axial two-body currents.
The vector current is related to the electromagnetic MEC operator that
contributes to electron scattering. This allows one to check our model by
comparison with the results of De Pace {\em et al.,} Nuclear Physics A 726
(2003) 303. Thus our model is a natural extension of that model to the weak
sector with the addition of the axial MEC operator. The dependences of the
response functions on several ingredients of the approach are analyzed.
Specifically we discuss relativistic effects, quantify the size of the
direct-exchange interferences, and the relative importance of the axial versus
vector current.
