The symmetric group $\mathfrak{S}_n$ can be regarded as a subgroup of the orthogonal group $\textrm{O}(n)$ via the permutation matrices. Let $V$ be a finite dimensional $\textrm{O}(n)$-module and $\varphi: \mathbb{R}^n\to V$ an $\mathfrak{S}_n$-equivariant linear map where $\mathfrak{S}_n$ acts on $\mathbb{R}^n$ in the obvious way. Finally, let $d:\mathbb{R}^n\to\textrm{Sym}_2(\mathbb{R}^n)$ be the map that sends a vector to the corresponding diagonal matrix.

Are there criteria on $\varphi$ that ensure the existence of an $\textrm{O}(n)$-equivariant linear map $\Phi: \textrm{Sym}_2(\mathbb{R}^n)\to V$ such that $\Phi\circ d=\varphi$?

Note that such a map, if it exists, is unique since every real symmetric matrix is diagonalizable with orthogonal matrices.

The symmetric group $\mathfrak{S}_n$ can be regarded as a subgroup of the orthogonal group $\textrm{O}(n)$ via the permutation matrices. Let $V$ be a finite dimensional $\textrm{O}(n)$-module and $\varphi: \mathbb{R}^n\to V$ an $\mathfrak{S}_n$-equivariant linear map where $\mathfrak{S}_n$ acts on $\mathbb{R}^n$ in the obvious way. Finally, let $d:\mathbb{R}^n\to\textrm{Sym}_2(\mathbb{R}^n)$ be the map that sends a vector to the corresponding diagonal matrix.

Are there criteria on $\varphi$ that ensure the existence of an $\textrm{O}(n)$-equivariant linear map $\Phi: \textrm{Sym}_2(\mathbb{R}^n)\to V$ such that $\Phi\circ d=\varphi$?

Note that such a map, if it exists, is unique since every real symmetric matrix is diagonalizable with orthogonal matrices.

Edit: This is to show that the map suggested by Aurel is in general not linear. Let $V$ be the representation of $\textrm{O}(n)$ where rotation about angle $t$ acts by $\begin{pmatrix}\cos(6t)&-\sin(6t)\\ \sin(6t)& \cos(6t)\end{pmatrix}$ and reflection at the $x$-axes by $\begin{pmatrix}1&0\\ 0& -1\end{pmatrix}$. Then one can check that the map $\varphi:\mathbb{R}^2\to V$ with $\varphi\binom{1}{1}=0$ and $\varphi\binom{1}{-1}=\binom{1}{0}$ satisfies the conditions described by Aurel. However, $V$ is not part of the decompostion of $\textrm{Sym}_2\mathbb{R}^2$ into irreducibles.

The symmetric group $\mathfrak{S}_n$ can be regarded as a subgroup of the orthogonal group $\textrm{O}(n)$ via the permutation matrices. Let $V$ be a finite dimensional $\textrm{O}(n)$-module and $\varphi: \mathbb{R}^n\to V$ an $\mathfrak{S}_n$-invariant linear map where $\mathfrak{S}_n$ acts on $\mathbb{R}^n$ in the obvious way. Finally, let $d:\mathbb{R}^n\to\textrm{Sym}_2(\mathbb{R}^n)$ be the map that sends a vector to the corresponding diagonal matrix.

Are there criteria on $\varphi$ that ensure the existence of an $\textrm{O}(n)$-invariant linear map $\Phi: \textrm{Sym}_2(\mathbb{R}^n)\to V$ such that $\Phi\circ d=\varphi$?

Note that such a map, if it exists, is unique since every real symmetric matrix is diagonalizable with orthogonal matrices.

The symmetric group $\mathfrak{S}_n$ can be regarded as a subgroup of the orthogonal group $\textrm{O}(n)$ via the permutation matrices. Let $V$ be a finite dimensional $\textrm{O}(n)$-module and $\varphi: \mathbb{R}^n\to V$ an $\mathfrak{S}_n$-equivariant linear map where $\mathfrak{S}_n$ acts on $\mathbb{R}^n$ in the obvious way. Finally, let $d:\mathbb{R}^n\to\textrm{Sym}_2(\mathbb{R}^n)$ be the map that sends a vector to the corresponding diagonal matrix.

Are there criteria on $\varphi$ that ensure the existence of an $\textrm{O}(n)$-equivariant linear map $\Phi: \textrm{Sym}_2(\mathbb{R}^n)\to V$ such that $\Phi\circ d=\varphi$?

Note that such a map, if it exists, is unique since every real symmetric matrix is diagonalizable with orthogonal matrices.