Suppose
$m$
and
$n$
are integers such that
$1\,\le \,m\,\le \,n$
. For a subgroup
$H$
of the symmetric group
${{S}_{m}}$
of degree
$m$
, consider the generalized matrix function on
$m\,\times \,m$
matrices
$B\,=\,\left( {{b}_{ij}} \right)$
defined by
${{d}^{H}}\left( B \right)\,=\,\sum{_{\sigma \in H}\,\prod{_{j=1}^{m}}\,{{b}_{j\sigma \left( j \right)}}}$
and the generalized numerical range of an
$n\,\times \,n$
complex matrix
$A$
associated with
${{d}^{H}}$
defined by
$${{W}^{H}}\left( A \right)\,=\,\left\{ {{d}^{H}}\left( {{X}^{*}}\,AX \right)\,:\,X\,is\,n\,\times \,m\,\text{such}\,\text{that}\,{{X}^{*}}X\,\text{=}\,{{\text{I}}_{m}} \right\}$$
It is known that
${{W}^{H}}\left( A \right)$
is convex if
$m\,=\,1$
or if
$m\,=\,n\,=\,2$
. We show that there exist normal matrices
$A$
for which
${{W}^{H}}\left( A \right)$
is not convex if
$3\,\le \,m\,\le \,n$
. Moreover, for
$m\,=\,2\,<\,n$
, we prove that a normal matrix
$A$
with eigenvalues lying on a straight line has convex
${{W}^{H}}\left( A \right)$
if and only if
$vA$
is Hermitian for some nonzero
$v\,\in \,\mathbb{C}$
. These results extend those of Hu, Hurley and Tam, who studied the special case when
$2\,\le \,m\,\le \,3\,\le \,n$
and
$H\,=\,{{S}_{m}}$
.