  We determine the superfluid transition temperature $T_c$ and related finite
temperature phase diagrams for the entire BCS-Bose Einstein condensation
crossover in a homogeneous mixture of $^{6}$Li and $^{40}$K atoms with
population imbalance. Our work is motivated by the recent observation of an
inter-species Feshbach resonance. Pairing fluctuation effects, which
significantly reduce $T_c$ from the onset temperature for pairing ($T^*$),
provide reasonable estimates of $T_c$ and indicate that the inter-species
superfluid phase should be accessible in future experiments. Although a
generalized-Sarma phase is not stable in the ground state near unitarity, our
phase diagrams show that it appears as an intermediate-temperature superfluid.
