Let $(V, \rho)$ be a representation of a finite group $G$ whose irreducible representations over complex numbers are $(W_i, \pi_i)$ for $1 \le i \le m$.
Suppose $\dim(\text{Hom}_G(W_i),V)=d_i$ and $\dim(W_i)=n_i$. We know $$V \cong \bigoplus_{i=1}^m W_i^{d_i}.$$ That is, $V$ can be identified by the direct sum of $m$ subspaces each of which is a $d_i$ copy of $W_i$. Also, according to https://math.stackexchange.com/q/4782272, $$\text{End}_G(V) \cong \bigoplus_i \text{End}_G(W_i^{d_i}) \cong \bigoplus_i M_{d_i}(\mathbb{C}). \ \ \ \ \ \ \ (*)$$
Based on this, I can identify elements of $\text{End}_G(V)$ by $A= \bigoplus_i A_i$ with some $d_i \times d_i$ matrices $A_i$. $A$ is a $\sum_i d_i \times \sum_i d_i$ matrix. However, since $\dim(V)= \sum_i n_id_i$, I would like to identify elements of $\text{End}_G(V)$ by $\sum_i d_in_i \times \sum_i d_in_i$ matrices.
I know there exists a basis of $V$, called symmetry-adapted basis, such that elements of $\text{End}_G(V)$ with respect to that basis have the form $\bigoplus_i I_{n_i} \otimes A_i$ for some $d_i \times d_i$ matrices $A_i$. The proof of this is elementary and based on Schur's lemma. However, I cannot make a relation between (*) and block diagonalization of elements of $\text{End}_G(V)$ with the symmetry-adapted basis.
Question: I expect to have $$\text{End}_G(W_i^{d_i}) \cong \mathbb{C}^{n_i} \otimes M_{d_i}(\mathbb{C}),$$ or $$\text{End}_G(W_i^{d_i}) \cong (M_{d_i}(\mathbb{C}))^{n_i},$$ but, according to (*), we have $$\text{End}_G(W_i^{d_i}) \cong \ M_{d_i}(\mathbb{C}).$$
Is my expectation true in some ways?
What should I do to make a relation between (*) and matrices of $\text{End}_G(V)$ with respect to the symmetry-adapted basis?