Finite groups with some restriction on the vanishing set

Let $x$ be an element of a finite group $G$ and denote the order of $x$ by $\mathrm{ord}(x)$. We consider a finite group $G$ such that $\gcd(\mathrm{ord}(x),\mathrm{ord}(y))\leqslant 2$ for any two vanishing elements $x$ and $y$ contained in distinct conjugacy classes. We show that such a group $G$ is solvable. When $G$ with the property above is supersolvable, we show that $G$ has a normal metabelian $2$-complement.