  Let $\mathfrak{a}$ be an ideal of a commutative noetherian ring $R$,
$\mathcal{S}$ a Serre subcategory of $R$-modules satisfying the condition
$C_\mathfrak{a}$ and $\mathcal{N}$ the subcategory of finitely generated
$R$-modules. In this paper, we continue the study of
$\mathcal{NS}$-$\mathfrak{a}$-cofinite modules with respect to the extension
subcategory $\mathcal{NS}$, show that some classical results of
$\mathfrak{a}$-cofiniteness hold for $\mathcal{NS}$-$\mathfrak{a}$-cofiniteness
in the cases $\mathrm{dim}R=d$ or $\mathrm{dim}R/\mathfrak{a}=d-1$, where $d$
is a positive integer. We also study $\mathcal{NS}$-$\mathfrak{a}$-cofiniteness
of local cohomology modules and the modules $\mathrm{Ext}^i_R(N,M)$ and
$\mathrm{Tor}_i^R(N,M)$.
