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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 $h'\mapsto h'^{-1},h'\in H'$.