Let $(W, \pi)$ be an irreducible representation of a finite group $G$ over complex numbers and $\rho: G \to GL_{N}(\mathbb{C})$ such that there exists $G$-isomorphism $\theta$ from $\mathbb{C}^N$ onto $W^d$. Suppose $\text{End}_G(\mathbb{C}^N)=\{A \in \text{Mat}_N(\mathbb{C}) \mid \rho_gA=A\rho_g\ \ \ \ \forall g \in G\}$.
Consider $W^d=W_1\oplus \cdots \oplus W_d$, where $W_i=W$, and consider $\pi_i:W^d \to W_i$ and $\eta_i:W_i \to W^d$ to be the natural projection and injection for each $i$. One can show that $$\tilde{\theta}: \text{End}_G(\mathbb{C}^N) \to \text{Mat}_d(\text{End}_G(W))$$ given by $\tau \mapsto (\tau_{ij})$ where $\tau_{ij}= \pi_i \circ \theta \circ \tau \circ \theta^{-1} \circ \eta_j$ is an algebra isomorphism. Also, by Schur's lemma, we can identify $\text{End}_G(W)$ by $ \mathbb{C}$. So, let say $$\tilde{\theta}: \text{End}_G(\mathbb{C}^N) \to \text{Mat}_d(\mathbb{C})$$ is an isomorphism.
Can we say that, for every symmetric, Hermitian, or positive semidefinite matrix $A \in \text{End}_G(\mathbb{C}^N)$, $\tilde{\theta}(A)$ is symmetric, Hermitian, or positive semidefinite matrix? If yes, how can we prove it?