Exact estimates of approximation by abstract Kantorovich type operators in terms of the second modulus of continuity

Approximation of bounded measurable functions on the segment $[0, 1]$ by Kantorovich type operators
$$ B_n(f)(x)=\sum_{j=0}^nC_n^jx^j(1-x)^{n-j}F_{j}(f), $$
where $F_{j}$ are functionals possessing sufficiently small supports and having some symmetry is considered. The error of approximation is estimated in terms of the second modulus of continuity. The result is sharp.