Let $G$ be a finite group, $k = \mathbb C$ and consider the ring $R = k[G]$. We can think of $M = k[G]$ as a left $R$ module and by considering character theory, we can show that there is an isomorphism of $R$-modules $f: M \cong N = \bigoplus_i V_i^{n_i}$ where the $V_i$ are the irreducible representations of $G$ with dimension $n_i$.
Now, $End_R(M) = k[G]$ where we think of $g$ as acting on the right on $M$ under this isomorphism. On the other hand $End_R(N) = \prod_i M_{n_i}(k)$ by Schur's lemma. The isomorphism $f$ induces an isomorphism of algebras $\hat f: End_R(M) = k[G] \cong \prod_i M_{n_1}(k) = End_R(N)$ that sends an endomorphism $A \to fAf^{-1}$.
On the other hand, we also have an isomorphism: $$k[G] \to \prod_iM_{n_i}(k)$$ that sends $g \to (\rho_V(g))_g$ where $\rho_V$ is the representation associated to $V$.
Are these two isomorphisms the same? More precisely, can we choose a $f$ so that this above isomorphism is equal to $\hat f?$