  Two-color lattice QCD with N_f=4 staggered fermion degrees of freedom (no
rooting trick is applied) with equal electric charge q is studied in a
homogeneous magnetic background field B and at non-zero temperature T. In order
to circumvent renormalization as a function of the bare coupling we apply a
fixed-scale approach. We study the influence of the magnetic field on the
critical temperature. At rather small pseudo-scalar meson mass ($m_{\pi}
\approx 175 \mathrm{MeV} \approx T_c(B=0)$) we confirm a monotonic rise of the
quark condensate $<\bar{\psi} \psi>$ with increasing magnetic field strength,
i.e. magnetic catalysis, as long as one is staying within the confinement or
deconfinement phase. In the transition region we find indications for a
non-monotonic behavior of $T_c(B)$ at low magnetic field strength ($qB<0.8
\mathrm{GeV}^2$) and a clear rise at stronger magnetic field. The conjectured
existence of a minimum value $T_c(B^{*}) < T_c(B=0)$ would leave a temperature
window for a decrease of $<\bar{\psi} \psi>$ with rising $B$ (inverse magnetic
catalysis) also in the present model.
