  Let $B$ be a finite dimensional algebra and $A=B[P_0]$ be the one-point
extension algebra of $B$ with respect to the finitely generated projective
$B$-module $P_0$. The categories of $B$-modules and $A$-modules are related by
two adjoint functors $\mathcal{R}$ and $\mathcal{E}$, called the restriction
and the extension functors, respectively. Based on the nice homological
properties of these two functors, restriction and extension of some notions
such as tilting and $\tau$-tilting modules have been studied in the category of
finitely presented modules, i.e. small mod. In this paper, we investigate the
behaviour of tilting and support $\tau$-tilting subcategories with respect to
these two functors. Moreover, we investigate the restriction and the extension
of special related modules such as finendo quasi-tilting modules, silting
modules, and cosilting modules. Our studies will be done in the category of all
modules, which will be called large Mod. Based on such study, in addition to
the new results, classical results are extended, not only from modules to
subcategories but also from small mod to large Mod.
