The fundamental group functor is not full. Counterexample? Subcategories with full restriction?

Anyone aware of a nice counterexample to "The fundamental group functor is full?" (Which is...false, right?) and are there a nontrivial subcategories on which its restriction is full?

I.e. Can you think of an example of topological spaces $X$ and $Y$ such that there is a group homomorphism $\phi: \pi_1(X, x_0) \to \pi_1 (Y,y_0)$ such that $\phi$ is not induced by any continuous map $f: X \to Y$ ?

Sometimes every homomorphism is induced by a continuous map: For example every automorphism of $\pi_1(S^1)$ is induced by a map $S^1 \to S^1$. Are all automorphisms of fundamental groups induced by continuous maps? What conditions on spaces $X$ and $Y$ ensure every fundamental group homomorphism is induced by a continuous map?

Solution 1:

Consider $X = \mathbb R P^3$ and $Y = \mathbb R P^2$. They both have $\pi_1$ equal to $\mathbb Z/2\mathbb Z$, but I don't think there is any continuous map $X \to Y$ which induces the identity on $\pi_1$s.