Induced $G$-spaces and induced $G$-$C^*$-algebras

Question

I have some trouble understanding 1.6.1(2) at the end. If I start with a locally compact Hausdorff $H$-space $Y$, why would the induced $G$-space be locally compact and Hausdorff? This has to be the case to make sense, doesn't it? I think the $G$-action needs to be proper for the induced space to be locally compact and Hausdorff.

Thank you