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In this short note we prove that an involution on certain examples ofsurfaces of general type with $p_g=0$, acts as identity on the Chow group ofzero cycles of the relevant surface. In particular we consider examples of suchsurfaces when the quotient is an Enriques surface and show that the Bloch'sconjecture holds for such surfaces.

Bloch's conjecture on certain surfaces of general type with $p_g=0$ and with an involution: the Enriques case