  Let $G$ be a finite group and $H$ a normal subgroup of prime index $p$. Let
$V$ be an irreducible ${\mathbb F}H$-module and $U$ a quotient of the induced
${\mathbb F}G$-module $V\kern-3pt\uparrow$. We describe the structure of $U$,
which is semisimple when ${\rm char}({\mathbb F})\ne p$ and uniserial if ${\rm
char}({\mathbb F})=p$. Furthermore, we describe the division rings arising as
endomorphism algebras of the simple components of $U$. We use techniques from
noncommutative ring theory to study ${\rm End}_{{\mathbb
F}G}(V\kern-3pt\uparrow)$ and relate the right ideal structure of ${\rm
End}_{{\mathbb F}G}(V\kern-3pt\uparrow)$ to the submodule structure of
$V\kern-3pt\uparrow$.
