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 Abstract: A {\it pseudocharacter\/} of a semigroup $S$ is a real function $\varphi$ on $S$ satisfying the following conditions. The set $\{\varphi(xy)-\varphi(x)-\varphi(y) ;\;\; x,y\in S\}$ is bounded. For $x\in S$ and $n\in \N$ (and $n\in \Z$ if $S$ is a group), $\varphi(x^n)=n\varphi(x).$ A description of the space of pseudocharacters on some extensions of free groups is given. Author information: Tver State Agricultural Academy, Tver Sakharovo, Russia Valeriy.Fayziev@tversu.ru