  Let $G,H$ be groups, $\phi: G \rightarrow H$ a group morphism, and $A$ a
$G$-graded algebra. The morphism $\phi$ induces an $H$-grading on $A$, and on
any $G$-graded $A$-module, which thus becomes an $H$-graded $A$-module. Given
an injective $G$-graded $A$-module, we give bounds for its injective dimension
when seen as $H$-graded $A$-module. Following ideas by Van den Bergh, we give
an application of our results to the stability of dualizing complexes through
change of grading.
