Let $(V, \rho)$ be a representation of a finite group $G$ whose irreducible representations over complex numbers are $(W_i, \pi_i)$ for $1 \le i \le m$. Let $$H_i=\text{Hom}_G(W_i,V)= \{\tau \in \text{Hom}(W_i,V) \mid \tau \pi_i(g)=\rho(g) \tau\ \ \ \ \forall g \in G\}.$$
Question: What are the rank and range of elements of $\text{Hom}_G(H_i,V)$?
What is the dimension of $\text{Hom}_G(H_i,V)$?
Motivation
Let $d_i=\dim(H_i)$ and $n_i=\dim(W_i)$. From representation theory, we know $V \cong V_1 \oplus \cdots \oplus V_m$, where $V_i=W_i^{\oplus d_i}$. That is, $V$ can be identified by the direct sum of $m$ subspaces each of which is a $d_i$ copy of $W_i$.
Moreover, each $V_i \cong H_i^{\oplus n_i}$. That is, each $V_i$ is isomorphic to a $n_i$ copy of the subspace of $G$-linear map from $W_i$ to $V$.
Added later:
We can see this if we let G act on $H_i \otimes W_i$ through the tensor product of the trivial representation of G on $H_i$ and the given representation on $W_i$, then $\sum h \otimes w \mapsto \sum h(w)$ defines an isomorphism from $H_i \otimes W_i$ onto $V_i$. Now, if we choose a basis for $W_i$, then $V_i \cong H_i \otimes \mathbb{C}^{n_i}$.
So, $$\text{Hom}_G(V,V) \cong \text{Hom}_G( \bigoplus_{i=1}^m V_i,V) \cong \bigoplus_{i=1}^m \text{Hom}_G(V_i,V),$$ where $$\text{Hom}_G(V_i,V) \cong \text{Hom}_G(H_i^{\oplus n_i},V) \cong \text{Hom}_G(H_i,V)^{\oplus n_i}.$$
That is why I would like to know more about $\text{Hom}_G(H_i,V)$ and its elements.
Since $$\text{Hom}_G(V_i,V) \cong \text{Hom}_G(W_i,V)^{\oplus d_i} \cong \text{Hom}_G(H_i,V)^{\oplus n_i},$$ I suspesct $n_i \dim(\text{Hom}_G(H_i,V))= d_i^2$. But, this does not make sense since what if $n_i \nmid d_i^2$.
Here is the correct calculation. We have
$$\begin{eqnarray*} \text{Hom}_G(W_i, V) &\cong& \text{Hom}_G(W_i, \bigoplus_j W_j^{d_j}) \\ &\cong& \bigoplus_j \text{Hom}_G(W_i, W_j^{d_j}) \\ &\cong& \text{Hom}_G(W_i, W_i)^{d_i} \\ &\cong& \mathbb{C}^{d_i} \end{eqnarray*}$$
by the universal property of the direct sum, then Schur's lemma. This tells us that a homomorphism $W_i \to V$ must land in the $W_i$-isotypic component $W_i^{d_i}$, and since by Schur's lemma it must be either zero or injective, its rank is either zero or $\dim W_i$, and its range is either zero or some invariant subspace of $W_i^{d_i}$ isomorphic to $W_i$.
$\text{Hom}_G(W_i, V)$ calculates the "multiplicity space" of the irreducible representation $W_i$ in $V$, which is an invariant version of the multiplicity.
Similarly we have
$$\begin{eqnarray*} \text{End}_G(V) &\cong& \text{Hom}_G(\bigoplus_i W_i^{d_i}, \bigoplus_j W_j^{d_j}) \\ &\cong& \bigoplus_{i, j} \text{Hom}_G(W_i^{d_i}, W_j^{d_j}) \\ &\cong& \bigoplus_i \text{End}_G(W_i^{d_i}) \\ &\cong& \bigoplus_i M_{d_i}(\mathbb{C}) \end{eqnarray*}$$
again by the universal property of the direct sum, then Schur's lemma.