Let
$R$
be a commutative Noetherian ring,
$\mathfrak{a}$
an ideal of
$R$
and
$M$
a finitely generated
$R$
-module. Let
$t$
be a non-negative integer. It is known that if the local cohomology module
$\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$
is finitely generated for all
$i\,<\,t$
, then
$\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$
is finitely generated. In this paper it is shown that if
$\text{H}_{\mathfrak{a}}^{i}\,\left( M \right)$
is Artinian for all
$i\,<\,t$
, then
$\text{Ho}{{\text{m}}_{R}}\,\left( R/\mathfrak{a},\,\text{H}_{\mathfrak{a}}^{t}\,\left( M \right) \right)$
need not be Artinian, but it has a finitely generated submodule
$N$
such that
$\text{Ho}{{\text{m}}_{R}}\left( R/\mathfrak{a},\text{H}_{\mathfrak{a}}^{t}\left( M \right) \right)/N$
is Artinian.