Suppose $E$ is a connective spectrum, then there exists a natural map in the stable homotopy category $\mathcal{SHC}$, $E \rightarrow P_0 E$, called the $0$-th Postnikov truncation, which is characterized by the property of being an isomorphism in $\pi_0$ and killing all higher homotopy groups. See for example Schwede, chapter II, 8.
The zeroth-Postnikov truncation is (weakly equivalent to) an Eilenberg-MacLane spectrum $H \pi_0 E$. In either the category of symmetric or orthogonal spectra we do have an explicit model for $HA$ for an abelian group $A$ as $\tilde{A}( \mathbb{S})$ where $\tilde A$ is the reduced linearization functor.
The Postnikov truncation of the sphere spectrum for example is given by the Hurewicz map, which is a natural map of symmetric (or orthogonal) spectra $\mathbb S \rightarrow H \mathbb Z$, defined by sending $x \mapsto 1 \cdot x$.
Can we give in a similar manner for an arbitrary symmetric spectrum a natural map of symmetric spectra $E \rightarrow H \pi_0 E$ which realizes the $0$-th Postnikov truncation in the stable homotopy category? Can we do so with sufficient restrictions on $E$ (e.g. being cofibrant) or with different models of $H \pi_0 E$? (Or alternatively in orthogonal spectra)
