Nope

The answer is no. Take $H$ any topological group, $G=H\times\mathbb{Z}/2$, H'$ another topological group having a noncentral element $a$ the generator a'$ of order 2, $\mathbb{Z}/2$ G=H\times H'$, $a=1_H\times a'$ and $f=g\times id_{\mathbb{Z}/2}$ f=h\times inv_{H'}$ where $g$ h$ is any endomorphism anti-automorphism of $H$.H$ and $inv_{h'}$ is the map $h'\mapsto h'^{-1},h'\in H'$.

Nope. Take $H$ any topological group, $G=H\times\mathbb{Z}/2$, $a$ the generator of $\mathbb{Z}/2$ and $f=g\times id_{\mathbb{Z}/2}$ where $g$ is any endomorphism of $H$.