Pseudocharacters on free groups, invariant with respect to some types of endomorphisms

We investigate methods for constructing nontrivial pseudocharacters on free group $F_n$ invariant with respect to certain types of endomorphisms. We find some conditions for endomorphisms of the free group under which there is a nontrivial pseudocharacter that is invariant with respect to these endomorphisms. We consider free products $R=\tilde R*\prod_{i=k}^n\langle r_i \rangle$, where one factor is $F_n$, and the other factor is a group on which there is a pseudocharacter. For such products we obtain a similar result about the conditions of existence of nontrivial pseudocharacters invariant with respect to certain endomorphisms.