  We consider two flavor magnetized quark matter in the presence of a repulsive
vector coupling ($G_V$) devoting special attention to the low temperature
region of the phase diagram to show how this type of interaction
counterbalances the effects produced by a strong magnetic field. The most
important effects occur at intermediate and low temperatures affecting the
location of the critical end point as well as the region of first order chiral
transitions. When $G_V=0$ the presence of high magnetic fields ($eB \ge 10
m_\pi^2$) increases the density coexistence region with respect to the case
when $B$ and $G_V$ are absent while a decrease of this region is observed at
high $G_V$ values and vanishing magnetic fields. Another interesting aspect
observed at the low temperature region is that the usual decrease of the
coexistence chemical value (Inverse Magnetic Catalysis) at $G_V=0$ is highly
affected by the presence of the vector interaction which acts in the opposite
way. Our investigation also shows that the presence of a repulsive vector
interaction enhances the de Haas-van Alphen oscillations which, for very low
temperatures, take place at $eB \lesssim 6 m_\pi^2$. We observe that the
presence of a magnetic field, together with a repulsive vector interaction,
gives rise to a complex transition pattern since $B$ favors the appearance of
multiple solutions to the gap equation whereas $G_V$ turns some metastable
solutions into stable ones allowing for a cascade of transitions to occur.
