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.
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