The present paper represents a continuation of paper [4], in which the existence and uniqueness problems for a general Ito--Volterra integrodifferential equation are investigated by using the concept of a non-linear random integral contractor. Since the Lipschitz condition and the random integral contractor for the coefficients of the considered equation, in general, cannot be compared, the notions of the modified Lipschitz condition and modified integral contractor are introduced on some function spaces, as well as the conditions of their equivalence. Some existence and uniqueness theorems are also given.