You have to understand how $B$ acts on $\operatorname{Hom}_A(V_i, V)$ 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
$$\bigoplus_i V_i \otimes \operatorname{Hom}_A(V_i, V) \to V.$$
