  Let $S$ be a smooth projective surface with $p_g=0$, let $\iota $ be a
regular involution acting on $S$, and let $W$ be the resolution of
singularities of the quotient surface $S/\iota $. In the paper we prove that
Bloch's conjecture holds for the surface $S$ if and only if it holds for the
surface $W$. This yields Bloch's conjecture for all surfaces $S$ whenever the
same conjecture is true for the desingularized quotient $W$. In particular,
Bloch's conjecture holds true for all numerical Godeaux surfaces with
involutions, a "half" of Campedelli surfaces with involutions, the surface of
Craighero and Gattazzo, some Catanese surfaces and other examples. Applying the
same method to $K3$-surfaces, we prove that if a $K3$-surface $S$ admits a
regular involution whose quotient is of Enriques type, then the motive $M(S)$
is finite-dimensional.
