Multivalued sections and self-maps of sphere bundles

Let $G$ be a finite group and $V$ a finite dimensional (non-zero) orthogonal $G$-module such that, for each prime $p$ dividing the order of $G$, the subspace of $V$ fixed by a Sylow $p$-subgroup of $G$ is non-zero and, if the dimension of $V$ is odd, has dimension greater than $1$. Using ideas of Avvakumov, Karasev, Kudrya and Skopenkov and work of Noakes on self-maps of sphere bundles, we show that, for any principal $G$-bundle $P\to X$ over a compact ENR $X$, there exists a $G$-map from $P$ to the unit sphere $S(V)$ in $V$.