The fundamental group is a functor from the category of pointed topological spaces to the category of groups.
Therefore every base-point preserving continuous function $f$ between pointed topological spaces induces a homomorphism $f_*$ between the fundamental groups. This is done by composing the loops with $f$, which is well-defined, because homotopy is also preserved under $f$.
Can we switch this around?
Every group is the fundamental group of a CW-complex, which can be constructed according to how many generators and relations the group has.
Can a continuous function be constructed for every homomorphism such that the continuous function induces the homomorphism? If the fundamental group functor is 'surjective', one has a pre-image at least.
How do you go from the algebraic to the topological with the morphisms? I have no idea.