Consider the equation $x^2=x_0$ in the symmetric group $S_n$, where $x_0\in S_n$ is fixed. Is it true that for each integer $n\geq 0$, the maximal number of solutions (the number of square roots of $x_0$) is attained when $x_0$ is the identity permutation? How far may it be generalized?
$\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL}$
The maximum of the function counting square roots is attained at $x_0=1$ and this statement generalises quite well.

Let $s(\chi)$ denote the Frobenius-Schur indicator of the irreducible character $\chi$. For the definition, see the edit below. One has $s(\chi)=1$ if the representation of $\chi$ can be realised over $\mathbb{R}$, $s(\chi)=-1$ if $\chi$ is real-valued but the corresponding representation is not realisable over $\mathbb{R}$, and $s(\chi)=0$ if $\chi$ is not real-valued. Then, the number of square roots of an element $g$ in any group is equal to
$$\sum_\chi s(\chi)\chi(g),$$
where the sum runs over all irreducible characters of the group. See below for a reference and a quick proof of this identity.

It follows from the usual theory of representations of $S_n$ that in this special case all Frobenius-Schur indicators are $1$, so the number of square roots of $x_0$ is just $\sum_\chi \chi(x_0)$. This proves that the maximal number of solutions is indeed attained by $x_0 = 1$, since each character value attains its maximum there. This generalises immediately to all groups for which every representation is either realisable over $\mathbb{R}$ or has non-real character, in other words has no symplectic (or sometimes called quaternionic) representations. That includes all abelian groups, all alternating groups, all dihedral groups, $\GL_n(\mathbb{F}_q)$ for all $n\in \mathbb{Z}_{\geq 1}$ and all prime powers $q$ (see [1, Ch. III, 12.6]), and many more.

[1] A. Zelevinsky, Representations of Finite Classical Groups, Lecture Notes in Mathematics, Vol. 869, Springer-Verlag, New York/Berlin, 1981.



Edit: One reference I have found for the identity expressing the number of square roots in terms of Frobenius-Schur indicators is Eugene Wigner, American Journal of Mathematics Vol. 63, No. 1 (Jan., 1941), pp. 57-63, "On representations of certain finite groups". Once you get used to the notation, you will recognise it in displayed formula (11). Since the notation is really heavy going, I will supply a quick proof here:

Claim: If $n(g)$ is the number of square roots of an element $g$ of a finite group $G$, then we have
$$n(g) = \sum_\chi s(\chi)\chi(g),$$
where the sum runs over all characters of $G$, and $s(\chi)$ denotes the Frobenius-Schur indicator of $\chi$, defined as $s(\chi)=\frac{1}{|G|}\sum_{h\in G}\chi(h^2)$.

Proof: It is clear that $n(g)$ is a class function, so it is a linear combination of the irreducible characters of $G$. The coefficient of $\chi$ in this linear combination can be recovered as the inner products of $n$ with $\chi$. We can write $n(g) = \sum_h \delta_{g,h^2}$ (here $\delta$ is the usual Kronecker delta), so we obtain
$$
\begin{align*}
\left< n,\chi \right> &= \frac{1}{|G|}\sum_{g\in G}n(g)\chi(g) = \frac{1}{|G|}\sum_{g\in G}\sum_{h\in G}\delta_{g,h^2}\chi(g)=\\\\
 &=\frac{1}{|G|}\sum_{h\in G}\sum_{g\in G}\delta_{g,h^2}\chi(g) = \frac{1}{|G|}\sum_{h\in G}\chi(h^2),
\end{align*}$$
as claimed.



Edit 2: I got curious and ran a little experiment. The proof above applies to all finite groups that have no symplectic representations. So the natural question is: what happens for those that do? Among the groups of size $\leq 150$, there are 1911 groups that have a symplectic representation, and for 1675 of them, the square root counting function does not attain its maximum at the identity! There are several curious questions that suggest themselves: is there a similar (representation-theoretic?) 2-line criterion that singles out those 300-odd groups that satisfy the conclusion but not the assumptions of the above proof? What happens for the others? Can we find a complete characterisation of the groups whose square root counting functions is maximised by the identity? Following Pete's suggestion, I have started two follow-up questions on this business: one on square roots and one on $n$-th roots.
