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

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

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

$$\bigoplus_i V_i \otimes \operatorname{Hom}_A(V_i, V) \to V.$$

You have to understand how $B$ acts on $Hom_A(V_i, V)$ for an irreducible representation $V_i$ of $A$. This depends on your context.

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

$\oplus_i V_i \otimes Hom_A(V_i, V) \to V$.

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

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