  Let $(R,\fm)$ be a local ring and let $C$ be a semidualizing $R$--module. In
this paper, we are concerned in $C$--injective and $G_{C}$--injective
dimensions of certain local cohomology modules of $R$. Firstly, the injective
dimension of $C$ and the above quantities of dimensions is compared. Then, as
an application of the above comparisons, a characterization of a dualizing
module of $R$ is given. Finally, it is shown that if $R$ is Cohen-Macaulay of
dimension $d$ such that $\H_{\fm}^{d}(C)$ is $C$--injective, then $R$ is
Gorernstein. This is an answer to the question which was recently presented.
