  Let us consider the symmetric square transfer of the automorphic
representation $\pi$ associated to a modular form $f \in S_k(N,\epsilon)$. In
this article, we study the variation of the epsilon factor of
${\mathrm{sym}}^2(\pi)$ under twisting in terms of the local Weil-Deligne
representation at each prime $p$. As an application, we detect the possible
types of the symmetric square transfer of the local representation at $p$.
Furthermore, as the conductor of ${\mathrm{sym}}^2(\pi)$ is involved in the
variation number, we compute it in terms of $N$.
