This was asked at MSE but never answered.

Let G be a finite group and denote by$\,$ sq(G)$\,$ the number of squares in G$\,$, i.e. the number of elements in G which possess a square root. For example, if G is a group of odd order, $\;$ then$\qquad$ sq(G)$\, =\,$|G|$\;$since each element has a square root, while at the opposite extreme, sq(G)$\,$=$\,$1$\,$when G is an elementary abelian 2-group. For the symmetric groups, the values of sq(Sym(n)) are listed at $\,$https://oeis.org/A003483 $\,$. Finally, we note that both the dihedral and quaternion groups of order 8 share the value sq(G) = 2$\,$.

Questions: 1) Can $\,$sq(G)$\,$ be determined from information in the character table of G$\,$?

$\qquad$$\qquad$$\,$2) Is it true that $\,$sq(Sym(n))$\;$is divisible by every prime which is less than or equal to $\,$n$\,$ $\qquad$$\qquad$$\;$$\;$$\;$$\;$for $\,$n$\,$>$\,$3$\,$?

Thanks very much.