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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 $\varphi:G\to\mathbb{C}$.