  Let $R$ be a commutative Noetherian ring with non-zero identity,
$\mathfrak{a}$ an ideal of $R$, $M$ a finitely generated $R$--module, and $X$
an arbitrary $R$--module. In this paper, for non-negative integers $s, t$ and a
finitely generated $R$--module $N$, we study the membership of
$\operatorname{Ext}_{R}^{s}(N, \operatorname{H}^{t}_{\mathfrak{a}}(M, X))$ in
Serre subcategories of the category of $R$--modules and present some upper
bounds for the injective dimension and the Bass numbers of
$\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$. We also give some results on
cofiniteness and minimaxness of $\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$ and
finiteness of $\operatorname{Ass}_R(\operatorname{H}^{t}_{\mathfrak{a}}(M, X)$.
