  Let $F$ be a non Archimedean local field with odd residual characteristic,
and let $K$ be a hyperspecial maximal compact subgroup of the $p$-adic
symplectic group $G=\mathrm{Sp}_4(F)$. Let $\mathfrak{s}$ be an inertial class
for $G$ in the Bernstein decomposition of the category of smooth
representations of $G$, which is attached to a proper Levi subgroup $L$ of $G$.
  We prove that the $\mathfrak{s}$-typical irreducible representations of $K$
are the irreducible components of
$\mathrm{Ind}_{J_{\mathfrak{s}}}^{K}(\lambda_{\mathfrak{s}})$, where
$(J_{\mathfrak{s}},\lambda_{\mathfrak{s}})$ is an $\mathfrak{s}$-type for $G$
such that $J_{\mathfrak{s}}\subset K$, and
$(J_{\mathfrak{s}},\lambda_{\mathfrak{s}})$ is a $G$-cover of a Bushnell-Kutzko
maximal simple type for $L$.
