  In this paper, the structure of the nearly invariant subspaces for discrete
semigroups generated by several (even infinitely many) automorphisms of the
unit disc is described. As part of this work, the near $S^*$-invariance
property of the image space $C_\varphi({\rm ker\, } T)$ is explored for
composition operators $C_\varphi$, induced by inner functions $\varphi$, and
Toeplitz operators $T$. After that, the analysis of nearly invariant subspaces
for strongly continuous multiplication semigroups of isometries is developed
with a study of cyclic subspaces generated by a single Hardy class function.
These are characterised in terms of model spaces in all cases when the outer
factor is a product of an invertible function and a rational (not necessarily
invertible) function. Techniques used include the theory of Toeplitz kernels
and reproducing kernels.
