  [Background] The neutrino-nucleus ($\nu A$) cross section is a major source
of systematic uncertainty in neutrino-oscillation studies. A precise $\nu A$
scattering model, in which multinucleon effects are incorporated, is pivotal
for an accurate interpretation of the data. [Purpose] In $\nu A$ interactions,
meson-exchange currents (MECs) can induce two-nucleon ($2N$) knockout from the
target nucleus, resulting in a two-particle two-hole (2p2h) final state. They
also affect single nucleon ($1N$) knockout reactions, yielding a one-particle
one-hole (1p1h) final state. Both channels affect the inclusive strength. We
present a study of axial and vector, seagull and pion-in-flight currents in
muon-neutrino induced $1N$ and $2N$ knockout reactions on $^{12}$C. [Method]
Bound and emitted nucleons are described as Hartree-Fock wave functions. For
the vector MECs, the standard expressions are used. For the axial current,
three parameterizations are considered. The framework developed here allows for
a treatment of MECs and short-range correlations (SRCs). [Results] Results are
compared with electron-scattering data and with literature. The strengths of
the seagull, pion-in-flight and axial currents are studied separately and
double differential cross sections including MECs are compared with results
including SRCs. A comparison with MiniBooNE and T2K data is presented.
[Conclusions] In the 1p1h channel, the effects of the MECs tend to cancel each
other, resulting in a small effect on the double differential cross section.
$2N$ knockout processes provide a small contribution to the inclusive double
differential cross section, ranging from the $2N$ knockout threshold into the
dip region. A fair agreement with the MiniBooNE and T2K data is reached.
