Does there exists an example of an extension of topological groups $1 \to N \to E \to G \to 1$ admitting a section $s:G \to E$ which is continuous (or continuous in a neighbourhood of identity) and satisfy the propery that $s(x^{-1}) =s(x)^{-1}$. One can see that the extension $0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{S}^1 \to 1$ does not admit such a section.

For discrete groups there are many extensions admitting such a section. For example, consider the extension $0 \to \mathbb{Z} \stackrel{i}{\to} \mathbb{Z} \times \mathbb{Z}_2 \stackrel{p}{\to} \mathbb{Z}_4 \to 0$ with $i(x)= (2x, [x])$ and $p(x, [y])=[x + 2y]$. Clearly, the section $s$ defined by $s([0])=(0,[0])$, $s([1])=(-1,[1])$, $s([2])=(0,[1])$ and $s([3])=(1,[1])$ satisfy the property that $s(x^{-1})= s(x)^{-1}$. I am looking for an example in the setting of compact connected topological groups.