  Recently, we proved that the image of a Toeplitz kernel of dimension $>1$
under composition by an inner function is nearly $S^*$-invariant if and only if
the inner function is an automorphism. In this paper, we build on this work and
describe the minimal Toeplitz kernel containing the image of a Toeplitz kernel
under a composition operator with an inner symbol. Furthermore, this work is
extended to the minimal Toeplitz kernel containing the image of a Toeplitz
kernel under a weighted composition operator. In particular, the corresponding
cases for minimal model spaces are also formulated, generalizing known work on
the action of composition operators on model spaces. Finally, the equivalences
between Toeplitz kernels are used to formulate the specific maximal vectors for
several Toeplitz kernels with symbols expressed in terms of composition
operators and inner functions.
