Let $G$ be a finite group of even order which has only one non-principal irreducible character $\chi$ of degree $d$, $d\in \mathbb{N}$, with the following property (we name it $(*_d)$):
$(*_d)$: There exist $x,y\in G$ such that $o(x)=o(y)=2$ and $\chi(x)\neq \chi(y)$.
I know that $A_n$ ($n\geq 8$) has the property $(*_{n-1})$ and $S_n$ ($n\geq 4$) has the property $(*_1)$.
Now, for a suitable $d\in \mathbb{N}$, I have a question:
Is there any other solvable example of groups $G$ with the property $(*_d)$?