If I have a homomorphism $f: G\to H$ of groups, I get a homomorphism $R(f)\colon R(H)\to R(G)$ of representation rings in the opposite direction, by composition. Given two homomorphisms $f_1,f_2\colon G\to H$, it is sometimes the case that $R(f_1)=R(f_2)$. For example, this happens whenever $f_1$ and $f_2$ are conjugate, i.e. there exists $h\in H$ such that, for all $x\in G$, $f_1(x)=h f_2(x) h^{-1}$.

Question: Is it true that if $R(f_1)=R(f_2)$ then $f_1$ and $f_2$ are conjugate? (Or what's a nice counterexample?)

Closely related: Suppose that $f_1(x)$ is conjugate to $f_2(x)$ for each $x\in G$ (but the way they are conjugate *a priori* depends on $x$). Is $f_1$ in fact conjugate to $f_2$?

The questions make sense pretty generally, but I am mostly interested in, say, finite groups and complex representation rings, if it makes any difference.