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 $(\*\_d)$):



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> $(\*\_d)$: There exist $x,y\in G$ such that $o(x)=o(y)=2$ and $\chi(x)\neq \chi(y)$.
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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:



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> **Is there any other solvable example of groups $G$ with the property $(\*\_d)$?**
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Are there any groups $G$ with the property $(*_d)$?
I have found an example of a solvable group that has exactly one irreducible character of degree $27$, and that character takes on two different values on involutions. The group, which has order $2^6 3^3$, can be constructed as a semidirect product of an elementary abelian $2$-group $V$ of order $2^6$ acted on by a nonabelian $3$-group $T$ with order $3^3$ and exponent $3$.


To construct the group, observe that $T$ has an irreducible module $V$ of dimension $3$ over the field of order $4$. View $V$ as an elementary abelian $2$-group and construct the semidirect product. I checked that this works by building the group using the Magma software, and then asking Magma for the character table.


