Pseudocharacters on free semigroups invariant with respect to their automorphism groups

Let $\mathcal{F}$ be a free semigroup and let $A$ be an automorphism group of $\mathcal{F}$. A description is given of the space of real functions $\varphi$ on semigroup $\mathcal{F}$ satisfying the following conditions:

1) the set $\{\varphi (xy)-\varphi (x)-\varphi (y);\,\, x,y\in \mathcal{F} \}$ is bounded;

2) $\varphi (x^{n}) = n\varphi (x)$ for any $x\in \mathcal{F}$ and $n\in N$;

3) $\varphi (x^{\tau }) = \varphi (x) \quad \forall x\in \mathcal{F}$, and $\forall \tau \in A$.