Suppose $G$ is a finitely presented group with generators $a_1, \ldots, a_n$. Suppose $f \colon G \to G$ is a group endomorphism specified by defining $f(a_1), \ldots, f(a_n)$. As expected, we define a fixed point of $f$ to be any element $g \in G$ such that $f(g) = g$ and, as $f(\mathop{id}) = \mathop{id}$, we say that $\mathop{id}$ is the trivial fixed point.

For example, let $G = \langle a | \rangle$ and $f$ and $g$ be defined by $f(a) = \mathop{id}$ and $g(a) = a^2$. Note in both cases $f$ and $g$ have no non-trivial fixed points and for this particular group we can determine that an endomorphism $f$ has a non-trivial fixed point if and only if $f(a) = a$.

For what groups is it possible to determine whether or not any given endomorphism has a non-trivial fixed point?

I am particularly interested in the question of:

Is $\langle a, b, c | \rangle$ such a group?