It is a difficult problem to decide whether a real-valued character $\chi$ of a finite group $G$ can be afforded by a real representation.
This information is encoded in the Frobenius-Schur indicator $\epsilon(\chi)$ of $\chi$, which cannot be computed from the complex character table.
Suppose that $\chi$ lies in a $2$-block $B$. Gow and Murray have shown that the so-called extended defect group $E$ has a strong influence on $\epsilon(\chi)$. We put forward a precise formula for $\epsilon(\chi)$ when $B$ is nilpotent.