at.algebraic topology - Properties of the class of topological spaces possessing a CW-structure

Let ${mathcal C}$ be the class of topological spaces which carry a CW-structure (note that I do not want to fix some particular CW-structure).

Is it true that for a covering map $Estackrel{f}{to} B$ with $Ein{mathcal C}$ we have $Bin{mathcal C}$, too?

It is true that the total space of a covering lies in ${mathcal C}$ if the base space does, but the reverse implication is not clear to me.

Edit

As Algori pointed out, the quotient space is not even Hausdorff in general. What about finite regular coverings, i.e. those which come from a free action of a finite group on the total space? Is it true then that the quotient space carries a CW-structure, too?

I'm interested in that because this would imply that given a free group action of a finite group on a "nice" space like a CW-complex, one can always choose a CW-structure with respect to which $G$ just permutes cells. Then the corresponding cellular complex would be a (possibly nice) complex of ${mathbb Z}G$-modules (for example, if the space was a sphere, then this procedures can be used to construct a periodic ${mathbb Z}G$-resolution of the trivial module ${mathbb Z}$, showing that the group has to have periodic invariants like homology and cohomology; in this particular case, however, things behave well as the quotient space ${mathbb S}^n/G$ is still a compact manifold).

Thank you.