  In this article we study chiral symmetry breaking for quark matter in a
magnetic background, $\bm B$, at finite temperature and quark chemical
potential, $\mu$, making use of the Ginzburg-Landau effective action formalism.
As a microscopic model to compute the effective action we use the renormalized
quark-meson model. Our main goal is to study the evolution of the critical
endpoint, ${\cal CP}$, as a function of the magnetic field strength, and
investigate on the realization of inverse magnetic catalysis at finite chemical
potential. We find that the phase transition at zero chemical potential is
always of the second order; for small and intermediate values of $\bm B$,
${\cal CP}$ moves towards small $\mu$, while for larger $\bm B$ it moves
towards moderately larger values of $\mu$. Our results are in agreement with
the inverse magnetic catalysis scenario at finite chemical potential and not
too large values of the magnetic field, while at larger $\bm B$ direct magnetic
catalysis sets in.
