Certain quotient spaces are countably separated,III

Let G be a locally compact group, N a closed normal subgroup, and α a multiplier for G such that every α-representation of G is Type I and such that every α-representation of N is Type I. Let $\text{N}\text{?}{}^{\mathrm{<mtext>\alpha </mtext>}}$ be the α-dual space of N. Necessary and sufficient conditions are given such that $\text{N}\text{?}{}^{\mathrm{<mtext>\alpha </mtext>}}\text{G}$ is countably separated. Generalizations of many known sufficient conditions for countable separability are easy corollaries.