Expansion of normal subsets of odd-order elements in finite groups

Let $G$ be a finite group and $K$ a normal subset consisting of odd-order elements. The rational closure of $K$, denoted ${\mathbf{D}}_K$, is the set of elements $x\in G$ with the property that $⟨x⟩=⟨y⟩$ for some $y$ in $K$. If $K^2\subseteq {\mathbf{D}}_K$, we prove that $⟨K⟩$ is soluble.