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Two-color lattice QCD with N_f=4 staggered fermion degrees of freedom (norooting trick is applied) with equal electric charge q is studied in ahomogeneous magnetic background field B and at non-zero temperature T. In orderto circumvent renormalization as a function of the bare coupling we apply afixed-scale approach. We study the influence of the magnetic field on thecritical 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 thequark condensate $<\bar{\psi} \psi>$ with increasing magnetic field strength,i.e. magnetic catalysis, as long as one is staying within the confinement ordeconfinement phase. In the transition region we find indications for anon-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 conjecturedexistence of a minimum value $T_c(B^{*}) < T_c(B=0)$ would leave a temperaturewindow for a decrease of $<\bar{\psi} \psi>$ with rising $B$ (inverse magneticcatalysis) also in the present model.

Magnetic catalysis (and inverse catalysis) at finite temperature in two-color lattice QCD