  We give a classification of Fano threefolds $X$ with canonical Gorenstein
singularities such that $X$ possess a regular involution, which acts freely on
some smooth surface in $|-K_X|$, and the linear system $|-K_X|$ gives a
morphism which is not an embedding. From this classification one gets, in
particular, a description of some natural class of Fano--Enriques threefolds.
