You have to understand how $B$ acts on 
$\text{Hom}_A(V_i, V)$ for an irreducible representation $V_i$ of $A$. This depends on your context. 

If you can do that then the evaluation map induces an isomorphism

$\oplus_i V_i \otimes \text{Hom}_A(V_i, V) \to V$.