<p>We study the generation of chirped nonautonomous solitons on a nonzero continuous wave background in an inhomogeneous optical fiber exhibiting temporary varying parameters of group velocity dispersion, cubic nonlinearity, self-steepening, quintic nonlinearity, self-frequency shift, and gain/loss. A general similarity transformation which reduces the governing generalized nonlinear Schrödinger equation with time-varying coefficients to its constant-coefficient form is derived. The self-similar variables and constraints depicting the nonlinear parameter functions are determined. A rich variety of exact bright, gray and kink solitons on a continuous wave background is derived for the first time for the constant-coefficient model. We find that these soliton waveforms are characterized by a nonlinear chirp, which involves two intensity dependent contributions and depends crucially on the self-steepening and self-frequency shift parameters. We construct the analytical nonautonomous bright, gray and kink soliton solutions with nonvanishing amplitude for the variable coefficient equation describing the pulse transmission in the femtosecond regime. We discuss the dynamical evolution of the obtained chirped nonautonomous bright, gray and kink solitons in different soliton control systems with periodically and exponentially modulated dispersion. Adjusting group velocity dispersion and gain/loss coefficients produces diverse dynamical behaviors like snakelike, breathing, V-shaped, parabolic snakelike, and cubic snakelike propagation behaviors. The results will be useful for the further understanding of the transmission properties of optical localized pulses and may stimulate novel experiments in relevant research fields.</p>