  We study the one-loop gluon polarization tensor at zero and finite
temperature in the presence of a magnetic field, to extract the thermo-magnetic
evolution of the strong coupling $\alpha_s$. We analyze four distinct regimes,
to wit, the small and large field cases, both at zero and at high temperature.
From a renormalization group analysis we show that at zero temperature, either
for small or large magnetic fields, and for a fixed transferred momentum $Q^2$,
$\alpha_s$ grows with the field strength with respect to its vacuum value.
However, at high temperature and also for a fixed value of $Q^2$ we find two
different cases: When the magnetic field is even larger than the squared
temperature, $\alpha_s$ also grows with the field strength. On the contrary,
when the squared temperature is larger than the magnetic field, a turnover
behavior occurs and $\alpha_s$ decreases with the field strength. This
thermo-magnetic behavior of $\alpha_s$ can help explain the inverse magnetic
catalysis phenomenon found by lattice QCD calculations.
