  Being a wide band gap system GaMnN attracted considerable interest after the
discovery of highest reported ferromagnetic transition temperature $T_C$ $\sim$
940 K among all diluted magnetic semiconductors. Later it become a debate due
to the observation of either a ferromagnetic state with very low $T_C$ $\sim$ 8
K or sometimes no ferromagnetic state at all. We address these issues by
calculating the ferromagnetic window, $T_C$ Vs $p$, within a $t-t'$ Kondo
lattice model using a spin-fermion Monte-Carlo method on a simple cubic
lattice. We exploit the next-nearest-neighbor hopping $t'$ to tune the degree
of delocalization of the free carriers and show that carrier localization
(delocalization) significantly widen (shrunken) the ferromagnetic window with a
reduction (enhancement) of the optimum $T_C$. We connect our results with the
experimental findings and try to understand the ambiguities in ferromagnetism
in GaMnN.
