  We investigate non-autonomous solitons in a general coherently coupled
nonlinear Schr\"odinger (CCNLS) system with temporally modulated nonlinearities
and with an external harmonic oscillator potential. This general CCNLS system
encompasses three distinct types of CCNLS equations that describe the dynamics
of beam propagation in an inhomogeneous Kerr-like nonlinear optical medium for
different choices of nonlinear polarizations owing to the anisotropy of the
medium. We identify a generalized similarity transformation to relate the
considered model into the standard integrable homogeneous coupled nonlinear
evolution equations with constant nonlinearities, accompanied by a constraint
relation expressed in the form of the Riccati equation. With the help of a
non-standard Hirota's bilinearization method and exact soliton solutions, we
explore the impact of varying nonlinearities and refractive index in the
propagation and collisions analytically by reverse engineering. Interestingly,
we show the emergence of several modulated solitonic phenomena such as periodic
oscillation, amplification, compression, tunneling/cross-over, excitons, as
well as their combined effect in the single-soliton propagation and two-soliton
collisions with appropriate forms of nonlinearity. Notably, we identify a tool
to transform the nature of soliton collisions with certain type of
inhomogeneous nonlinearities. The results could be of significant interest to
the studies on management of nonlinear waves in the contexts like nonlinear
optics and can also be extended to Bose-Einstein condensates and super-fluids.
