Here's a counterexample: take $G=S^1=\{z\in\mathbb{C}:|z|=1\}$,
$$
f(z)=\begin{cases}z^2:\mathrm{Im}(z)\geq 0,\\
\overline{z}^2:\mathrm{Im}(z)\leq 0.
\end{cases}
$$
This $f$ is nullhomotopic, but is an odd map because $\int_G \varphi\circ f=\int_G \varphi$ for all $\varphi:G\to\mathbb{C}$.