Modules induced from a normal subgroup of prime index

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 $V\kern-3pt\uparrow$.