For a connected Lie group $G$ and its subgroup $H$, if $\pi_0(G/H) = 1$, is it true $\pi_0(\frac{G\times Z_2}{H}) = \{1,-1\}$ and $\pi_0(\frac{G\times Z_2}{H\times Z_2}) = 1$?
I have to understand something about the homotopy theory, but I don't have a mathematics background and don't know where I can find some theorem or explanation for the above question. Can anyone give me some hits? Either plain text explanation or introductory level references would be very appreciate!!!