Let C be the space of 2π-periodic continuons real functions with the uniform norm, let H_n be the set of trigonometric polynomials of order not more than n, let ω²(f) be the second continuity modulus for a function f ∈ C, and let T_n(f) be the best approximation polynomial of order n for f ∈ C. Set A₀(f) = 1/2π ∫-_π^π f; /; U : C - C; C(U,h) = sup _f∈C ||U(f) - ||/ω₂(f, h), In this paper, for h sufficiently large we find the values C(U,h) for some positive operators U. For example, C(A⁰, h) and C(T₀, h) are found. For n = 1, 2, 3 we find the values C(U, π/n + 1) for some linear positive operators U : C → H_n. We establish relations between C(T₀, h) and exact constants in the inequality ω₂(f, h₁) ≤ C(h₁, h)ω₂(f, h) for some h and h₁ such that 0 < h < h₁ < π. For a seminorm P invariant with respect to the shift and majorized by the uniform norm, analogs of C(U, h) are estimated from above. We investigate the problem of extension of a function defined on a segment with preservation of the second continuity modulus. The relation (Matrix equation presented) is established. Here the segment X contains I = [0, 1] as a proper subset, and ω₂ (f, X, h) is the second continuity modulus for f on X with step h. Bibliography: 5 titles.