  We perform a lattice study of the ($2+1$)-dimensional Gross-Neveu model in a
background magnetic field $B$ and at non-zero chemical potential $\mu$. The
complex-action problem arising in our simulations using overlap fermions is
under control. For $B=0$ we observe a first-order phase transition in $\mu$
even at non-vanishing temperatures. Our main finding, however, is that the rich
phase structure found in the limit of infinite flavor number $N_\mathrm{f}$ is
washed out by the fluctuations present at $N_\mathrm{f}=1$. We find no evidence
for inverse magnetic catalysis, i.e., the decrease of the order parameter of
chiral symmetry breaking with $B$ for $\mu$ close to the chiral phase
transition. Instead, the magnetic field tends to enhance the breakdown of
chiral symmetry for all values of $\mu$ below the transition. Moreover, we find
no trace of spatial inhomogeneities in the order parameter. We briefly comment
on the potential relevance of our results for QCD.
