Rigidity and rationality in finite groups

Let $G, H$ be finite groups with strongly rigid families

Show that then

is a strongly rigid family in $G \times H$.

Let $\sigma$ be an outer automorphism of a finite group, and let ($\mathcal{K}_{1}, \ldots , \mathcal{K}_{m}$) be a rigid family of conjugacy classes in $G$. Show that then $\sigma \left(\mathcal{K}_{i}\right) \neq \mathcal{K}_{i}$ for some $i$.

Show that in the alternating group $A_{5}$, with the conjugacy classes $1 A, 2 A, 3 A, 5 A, 5 B$ already known to us, the triples

are strongly rigid. It follows from this that $A_{5} \cong \mathrm{Gal} F / \mathbb {Q}(\sqrt{5})$.

Denote by $K_{1}(z)$ the number of representations of an element $z$ of a finite group $G$ as a commutator, i.e., the number of solutions of the equation $z=x y x^{-1} y^{-1}$. Show that

More generally: if $K_{m}(z)$ is the number of solutions of the equation

then