A nonautonomous linear system controlled by a nonlinear sector-restricted feedback with a time-varying delay is considered. Delay-independent sufficient conditions for absolute stability and instability (expressed in the transfer function of the linear part and the sector bounds) are established. For a system with an exponentially stable linear part, an upper bound for the Lyapunov exponent is found. It is shown that if the transfer function is sign-constant, asymptotic stability of the system with the margin-linear feedback guarantees absolute stability of the considered system; thus, such systems satisfy the known Aizerman conjecture. They include, in particular, a closed-loop system consisting of any number of time-varying first order links and feedback with arbitrary delay. Under some additional condition (which is certainly true for a time-invariant linear block), the obtained stability criterion is precise. The approach employed in the proofs is based on a direct analysis of the corresponding Volterra equation which contains only the transfer function of the linear block and, therefore, embraces a wide range of control systems. As an example, a second order system is considered; it is shown that here the obtained stability bound is reached for a linear feedback with a periodic delay function.