The number of solutions of equations $x^2=\sigma$ in symmetric groups

Question

Let $S_n$ be the symmetric group on $n$ letters, ant let $N(\sigma,n)$ be the number of solutions of the equation $x^2=\sigma$, $\sigma\in S_n$ in $S_n$.

Is it true that

For any $n\in\mathbb{N}$ and for any $\sigma\in S_n$ we have $N(e,n)\ge N(\sigma,n)$.

(where $e\in S_n$ is the trivial permutation).